Light kit ballast replacement
Ballast replacement is fairly straightforward.   I still prefer fluorescent tubes because I can easily find quality fluorescent tubes in 90+ CRI (in fact Menards stocks them), and I have serious doubts about the real quality and longevity of the LED "cheapie" tubes.  

Start with a "ballast disconnect"



For this you need to shut off the circuit breaker supplying power.  You then put the ballast disconnect at the point where the power supply (typically 12-14 AWG wire) meets the ballast (18 AWG wire).  Some disconnects care which side is the heavier wire, i.e. the yellow Wago types.  If there are 2 power cables going in and out, then either use the Wagos (which provide for that) or pigtail it.

Once that's done, you'll be able to service the fixture without turning the breaker off. Just unplug. 

Next select the bulbs

Those look like 48" bi-pin tubes (the most common one). From looking at the "old" Keystone ballast, your old ballast was for F32T8 tubes, 1" diameter (T8).  Which means it's not that old.  Don't use F40T12 tubes, 1-1/2" diameter, they will fit but they are not electrically compatible unless you change to a T12 ballast. 

Look at the existing ballast

The big label on the old Keystone ballast says your bulbs are instant-start, with 1 wire per end of the lamp.  Gotta watch out for that 2-yellow 2-red 2-blue color scheme as it's also used by 2-lamp rapid-start ballasts.  This is an example where the diagram really matters. 

Check the new ballast also, the wiring may be subtly different.  If the Lowes clerk picked your matching ballast by the same wiring diagram, that's handy, but double-check the type of tube it's made for.  Here are generic drawings of ballast wiring; yours may vary.  For instance yours wires like two separate 2-tube instant-start ballasts. 


This is from my big answer here. 

Historically 4-lamp fixtures often had 2 ballasts.  There is a mounting site in your fixture for a second ballast.  Aside from using the more common 2-lamp ballast. it also let you feed them from 2 switches for "high" and "low". 

If the ballast is a goner, you'll have no trouble finding 4-lamp ballasts in either instant (1 wire) or rapid-start (2 wires).  Rapid-start gives better tube life; a sub-type is programmed-start which gives superb tube life - something to think about if the fixture is hard to access.  But if you switch to rapid-start, you'll need to change the lampholders (tombstones) on the end of the tubes to a non-shorting type. 

Edited significantly  based on updated info