Why won't my car turn on?

Monday morning (3 days ago) I drove my car to the store and back. In the afternoon I went to turn it on to go to work only to get it to not start at all. It simply makes one click sound when I turn the key.
Here is some info:
1) I changed the battery to see if that was the cause but it's not
2) I was planning to take my car to Kwik Kar to get my oils checked and drained but I went to my car two hours before going to work to check the radiator to see if I was low on coolant. And I low on coolant/anti freeze. I told myself I was going to get it drained and refilled on Sunday as well.
3) Engine light has been on prior to this incident. The code was p0740. I don't have the money currently to fix it because I was told it would be costly.
4) I had not had problems turning it on prior to this happening. It really shocked me that it just died on me out of the blue without symptoms.

The car:
2002 Honda Civic ex (1.7 liter/4cyl) automatic

Here is a clip I uploaded of what the click sounds like when I try to turn on my car:


I was told by my coworkers that it may be the starter. I can't find a mechanic in my area that's willing to come to my house or some have offered but by charging me for my arm and leg.

What do you guys think? Any advice?

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