To make it clear from the start, this piece will not cover the current events in any detail. I'm writing this in an attempt to help explain the history of the major players in the current situation in Egypt and why the situation is so polarized. I am neither a journalist nor a historian, so please forgive any errors (if you could point them out, politely, and explain where I went wrong please do so in the comments). I will try to be as impartial as possible.
The first thing to understand is that both the Armed Forces and the Muslim Brotherhood are deeply embedded in Egypt's modern history and social conscious.
The Muslim Brotherhood was formed in 1928 by school teacher Hasan Al Bana, with the aim of returning to a Caliphite stretching from Spain to Indonesia, taking the Quraan and Sunnah (sharia law) as the guide to politics and society. For most of its history the brotherhood has been officially banned from politics and outlawed as an organisation. The first time came after the assassination of the Egyptian Prime Minister and the second under President Nasser after an attempt on his life. Despite having supported the 1952 revolution that removed the monarchy from power, they disagreed with the secular goals of the new regime.
The organization has gained support manly through charitable work ( especially in the form of food and healthcare to the needy) and because they were the most prominent form of opposition. This may help explain the loyalty at a grass roots level that the Brotherhood has amongst people, especially the poor. Where state institutions failed them, especially on an economic level, they were there to help cover bills and provide help. And of course, there's no one so willing to join an organization in opposition than those already on the fringes of society where it provides a sense of belonging and importance.
At the beginning of the uprising in 2011, their leaders asked their members not to join any of the protests, however as the movement became more popular and it looked like the police might retreat they joined in. This was enough to give their candidate the edge in the elections; despite having said that they would not be putting forward a candidate.
People have always been incredibly distrustful of them, despite this enough people decided to give Morsi a chance over Ahmed Shafiq who had been PM under Mubarak, to give him a slim margin to win the elections despite an incredibly low voter turnout in the second round.
Then came all the broken promises: a Copt as Vice President, plenty of cabinet positions for women, cracking down on the interior ministry and police brutality, to name a few. Instead more members of the Brotherhood and islamist parties were placed in key ministerial positions.The Brotherhood's political wing The Freedom and Justice Party ( FJP) then flooded the ballots ( along with more extreme "Islamists"). All their campaigns around poorer neighbourhoods involved bags of basic food stuffs and they were everywhere.
Parliament was elected and promptly set to work, their primary concerns seemed to be lowering the legal marriage age (to 12!), overturning the ban on female gentile mutilation and not teaching foreign languages in schools as they are 'corrupt'. Nothing about increasing wages, manufacture or any plan for the economy apart from trying to secure an IMF loan.
Then came the constitutional declaration in November which gave the President sweeping powers and was full of enough holes that even someone,like me, who struggles with legal jargon could see problems down the road. That lead to protests and a sit in at the presidential Palace which was forcibly broken up and people died ( not by the security forces but by supporters of the President) to no official condemnation.
That's not counting the things that made daily life, if not unbearable for all, then certainly inconvenient, daily power cuts and water shortages, garbage in the streets.
The police weren't exactly helping during all this time, as a kind of unofficial protest to having some of their powers curtailed and their torture brought to a wider light, they stopped doing their jobs. Crime rates escalated.
So, when a group of people decided to start the 'Tamarod' or Rebel campaign to call for early elections they ended up gathering more signatures than Morsi had votes to win the election (the movement claims at least 20 million signatures). Security forces saw this as a chance to gain back their previous positions of power and get the all important public opinion on their side and they promised to do whatever the people wanted and that they would protect the anti-Morsi demonstrations on June 30.
Egyptians love their Armed Forces. This dates back to 1952 when the Free Officers Movement lead a coup to overthrow the king- unlike recent events the king willingly abdicated to prevent bloodshed. Because of the British occupation, it was also a liberation from foreign rule and lead to a surge of patriotism and nationalism. The two main leaders were Mohamed Naguib and Gammal Abd El Nasser and while the latter is the most famous, Naguib was actually Egypt's first president, who due to disagreements with Nasser has been relegated to the footnotes of Egyptian history (if included at all).
Nasser on the other hand, is not a figure history can ignore. Love him or hate him, he was a charismatic leader and a gifted orator whose humble backgrounds meant the majority of Egyptians found him relatable. With proclaimed socialist leanings he insured that 50% of parliament seats went to workers and farmers as well as distributing farm land to those who worked it. He nationalised the Suez Canal which lead to the retaliatory invasion of the Sinai peninsula by Britain, France and Israel. After stepping down following the Six Day Way in 1964, he was reinstated by popular demand.
After his death came Anwar El Sadat who opened up the economy through trade agreements and signed the Camp David agreement. It was during this time that mass migration of the rural poor to the cities began (devastating the agricultural sector) and immigration to Saudi Arabia and the Gulf states.
For centuries Egyptians of all religions had lived peacefully side by side with neighbours celebrating religious and secular holidays together. There was a thriving and vital Jewish community. There was vibrant art scene and nightlife and through it all, Egyptians held fast to their faiths. Many claim that this began to change once people who had been working in Saudi Arabia began to return to Egypt with money and spreading the more conservative and fundamental Whabi form of Islam practiced there. Starting with the assassination of President Sadat in 1981, there began a crackdown on various armed islamist groups (several of which were off shoots of the Muslim Brotherhood). The 90s in particular were a violent time in Egypt, with bombs on government facilities, civilians and tourists all common. The perpetrators were the Islamists who looked 'different' and who seemed to be speaking of a different Islam than that which the majority of Egyptians practiced. It sought to reshape the country and it's people. They threatened security and the economy and so people supported any steps taken against them.
This may go some way to explain Egyptians' love and loyalty to the Armed Forces. They see them as having restored national pride and dignity to the people after decades of foreign rule. The police state whose end was called for in 2011 grew out of people's readiness to put down the enemies of the military and those they see as threatening Egypt's national identity and security.
What you see happening now is an -almost inevitable- implosion of the supporters of these two decade old foes.
Many of those opposed to Morsi's rule and who are supporting the current crackdown feel let down by the West, especially the media. For years Muslims around the world have been screaming that not all Muslim are like Al Qaeda and are trying to bring down Western civilization. The way they see it, is that what the security forces are doing now is clamping down on an organization that has bred terrorists who have effected the lives of millions around the world. They resent being told by the Brotherhood supporters that they are not 'true Muslims' for not wanting to live under their rule as they have always practiced the Muslim faith diligently. They feel that they are responding to what we hear so many Western politicians say whenever a Muslim commits a crime in the name of Islam; standing up to the extremists.
I hope that this has in some way improved your understanding of the current situation and why each side is as passionate as they are. Please also remember that your own democracies have been around for much longer and have already gone through most of these teething problems.