Will the light fixture be able to replace my current light fixture?
Should be fine. 

The old fixture has 2 hots, one going to each lamp socket.  They both go to the same place - a single "hot" supply wire.  That's where your new fixture's "hot" goes.  

You'll probably see something similar with the neutrals.  You might see 2 neutrals coming in from the ceiling and joining the 2 from the old lamp.  If so, join those two and the one from the lamp. 

The wires look ample for what the lamp is listed for -- 60 watt incandescents.  The reason for the restrictions is fire from the heat of those bulbs overheating the fixture.  So any bulb which consumes more than 60 actual watts is improper.  As far as CFLs, the fixture won't overheat from a hypothetical 60-watt-actual CFL (which would be bright), but I think they are trying to tell you the "bulb" would fail prematurely because the fixture is too confined for it too cool properly. (among other things CFLs are designed to run base-down not sideways).  And this is a guess by the fixture manufacturer.  

Since the year is 2017, I assume you'd use LEDs. They don't cool as well as CFLs, but have better heat sinks and much better efficiency in lumens per watt. You can also get directional LEDs which only light one "side" of the "bulb", so in a fixture like this you're not wasting energy lighting the top of the fixture.  So there's no problem getting plenty of light - for instance two 13W's directionals would be 3000 lumens of well-aimed light, that's as much as two 48" fluorescent tubes (rather quite a lot, and probably more than you want).

You were wise to take a "before" picture.  Many people fail to do this and wind up lost.