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?

The click you heard was the starter solenoid. The starter solenoid has internal electrical contacts which connects the battery to the starter motor. Internal solenoid contacts can burn off where they don't make contact and thus the starter motor gets no battery current to turn the starter motor. Back in the day it was possible to replace the inexpensive solenoid without replacing the expensive starter motor. It has been many decades since I repaired cars and you may not be able to replace just the solenoid today and have to replace the entire starter. You can still go the junkyard and buy a used starter for your Honda and replace it by yourself. In any event you need to visually inspect your starter looking and pulling for corroded or loose wires. That is what I would do before taking it to a mechanic.

It's probably the starter. I had to replace mine in a 2014 Honada
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IT DOES SOUND LIKE THE STARTER. BUT, I WOULD CHECK ALL OF THE CONNECTIONS BEFORE I PAID A WRECKER TO TOW IT.

In any 2002 Honda the top suspect is the ignition switch. (The electrical part is easily replaced.) Although the 2002 Civic escaped the recall, the ignition switch is the single most common failure in that era Honda and failure to engage the starter is the most common form of that failure.

How to know? You can get a remote starter switch at any auto parts store for about $20; the source is an example. Clip it onto the starter (see the instructions) and push the button. If the starter clicks: bad starter. If the starter cranks, bad ignition switch. I give you 2 to 1 it is the switch. BTW, if it is the switch you can use the remote starter switch to start your car. Put the ignition switch in the "run" position and use the remote starter to crank it. If you had a manual transmission you would have to be sure it was in neutral but with your automatic it will be in park anyway.

The code p0740 is a torque converter or clutch system malfunction. It sound like your transmission is locked, preventing the engine from turning over. It requires a professional to look into it, and it sounds expensive.

You say you changed the battery… To the one that was sitting out behind the shed or a brand new one? If this is a brand new battery I'd check to see if the engine turns over by hand before spending another dime. It sounds almost like the engine is seized.