OK, not sure how computer-literate you are, so let's start at the beginning, and please forgive me if I make it too simplified.
Everything black in the above picture is your Heat Sink Fan (HSF). The gray vertical lines behind it belong to your Heat Sink (HS). The HS draws heat away from your CPU to keep it cool and the HSF takes all that heat and blows it out. That view shows the top of your HS.
That view shows the bottom of your HS *and* the CPU. it's very common for the style Socket you have to pull the CPU out of it's Socket when you only want to remove the HS and HSF.
That is your Socket. The CPU should be locked into that, but again, if the thermal compound between the HS and CPU is old, it acts like a glue. So when you pulled off your HS, you pulled the CPU out with it. So...
All those 'little prongs' are part of the CPU, not the coolant fan. Your Socket (Picture 3 above) will have a lever on it's side. The lever will lock/unlock the CPU into place. Provided all the pins are there on your CPU, I would do this:
1: gently twist your CPU off the bottom of the HS
2: clean all the thermal compound from both the top of the CPU and the bottom of the HS
3: pull up the lever on the Socket
4: gently place the CPU into the Socket (it will only go one way - if it doesn't fit, turn it 1/4 way and try again - DO NOT force it)
5: push the lever on the Socket back down
6: apply a little bit of new thermal compound (use the Search feature on these forums to find out how) onto the top of the CPU
7: gently place the HS/HSF directly on top of the CPU (straight down, don't wiggle it)
8: lock the HS/HSF into place on the Mother Board.