Why did employees get fired before the big recession?

Because problems with the banks' finances were what caused the recession

The thing about the Great Recession is that ti was caused, to a large extent, by dodgy behaviour in banks. They made paper profits by growing their derivative businesses which were ultimately dependent on the growth of mortgage lending to people who could not really afford to pay back their mortgages. This was happening a long time before the recession hit.

The recession was triggered when the banks realised the underlying risk to their financial viability and started to try to reduce it.

The story of Margin call occurs at this early stage. A clever analyst has finally realised the existential risks they are facing and the bank chooses to take some dubious steps to fix the problem. But the bank's business would already have been seeing some of the problems before they realised that their very existence was at stake. Those early problems would have been dealt with by shrinking the cost base (well before the need to shed the liabilities that put the whole organisation at risk). Hence, layoffs.

The layoffs are because the normal profits were disappearing; the dodgy shenanigans that make up most of the story only occur when the bank realises that the consequences of that threatened the very existence of the bank.

It's called "Maximizing profits by minimalizing costs". Human costs are the easiest to cut and thus lower the amount of money the company is spending (that cannot be written off in taxes). Companies do that, see what people are really needed and then hire them back through temp companies or as "counsellor".