Your typical water to grain ratio is 1.33qts/lb. So I would assume your grain bill is ~ 14lbs. If not, just take your grain weight (in lbs) and multiply by 1.33 (or whatever ratio you would like. Higher ratios for thinner mashes, lower for stiff mash, etc)... So that is where the 19qts come in. That is called your strike water.
As Hunter_la5 stated, after your hour long mash, you will add the 0.89 gal of boil water. This will raise your overall mash temp up to roughly 160F. This is called your mash out. The purpose of the mashout is to denature the enzymes so you still arent converting sugar while sparging. You would then completely drain the mash tun, add the 4.10 gal of ~160F water, stir, and drain as well. This will net you your pre-boil volume
Now, if it were me, I would just combine the 0.89gal water addition and the 4.10gal water addition together. You can still achieve the mash out in a single batch sparge step.
If you are confused on why so much water is used, the grain will absorb a decent amount of the water during the mash. You will put in 19 qts, but you wont get near that out of your first runnings. Assuming you have little deadspace, you can expect to collect around 45% of your strike water volume. So that would mean your first running volume would be (19qt=4.75gal) a little over 2 gal (8.5qts). The ~5 gal you will batch sparge with wont be absorbed by the grain, as it has already met its max absorption. Therefore, you will collect an additional 5gal of wort from the sparge, netting you a pre-boil volume of 7gal, which is common for 5 gal. You will then boil this for an hour, add hops, etc, and at the end of it all, your post boil volume will be around 5.5gal (depending on boiloff rate, how vigorous the boil is, etc).
Transfer to the fermenter, (you will lose some volume, which is equipment loss), ferment, and package. In the end, you will net 5 gal of beer.