When you apply it has rationales, but it isn't really intuitive unless you're just really familiar with the rule set.
It boils down to each stat has 2 "modifiers" (the raw mod, and mod+stat), and it's extremely situational which one you use. (And of course the modifiers are different from the stat, but we won't go there.)
Are you attacking? Use the second one, except in weird cases. Are you rolling damage? Use the first one, never ever the second one.
Making a saving throw? Sometimes use the first, sometimes use the second, everyone is different and you better check your character sheet.
Making a skill check? Sometimes use the first, sometimes the second, everyone is different and you better check your character sheet. Oh, and a few characters have a special "third modifier" (expertise), so you definitely need to check your sheet for that so you don't forget.
Oh, and you're blessed? Well, definitely add a d4 to your attack roll. Your damage roll? Don't add the d4 there! Oh, you're being grappled? Don't add the d4 to that check! Oh wait, the grapple is part of a spell that has a save? Then yes, definitely add that d4! <sigh>