It's not really a checksum. It can sometimes function as one (everything should sum to zero, if it doesn't then you have a math error), but since
most records you make will just have two entries (spent $25 on groceries, remove $25 from checking account) the everything-sums-to-zero feature isn't going to catch math mistakes most of the time, because there is no math to be done on most entries. Rather, the fact that everything sums to zero helps you track things later on.
To explain that, I'll rephrase, in my own words, the restaurant example from the article, because that was a good example of the concept. Let's say you went to a restaurant with two friends and decided to split the $90 bill three ways, but your friends didn't have $30 in cash on them at the time. You put the whole $90 on your credit card, and your friends paid you back $30 each the next week: one on Monday, and one on Wednesday.
In single-entry accounting you might have written the following transactions:
Thursday Jan 1st: $90 restaurant (credit card)
Monday Jan 5th: $30 repaid from Alice (cash)
Wednesday Jan 7th: $30 repaid from Bob (cash)
Thing is, there's nothing to link those transactions together. If you look at these entries three years later, you'll probably be left scratching your head as to why Alice paid you back $30: there's no $30 transaction, so the $90 restaurant transaction won't jump out at you as the reason why Alice paid you back.
But with double-entry bookkeeping, you'd write that as follows:
Thursday Jan 1st:
-$90 restaurant (credit card)
+$30 my share of the restaurant bill (expenses)
+$30 Alice's share of the restaurant bill (money owed to me)
+$30 Bob's share of the restaurant bill (money owed to me)
Monday Jan 5th:
-$30 Alice's share of the restaurant bill (money owed to me)
+$30 cash received (cash)
Wednesday Jan 7th:
-$30 Bob's share of the restaurant bill (money owed to me)
+$30 cash received (cash)
It's not always obvious when you're new to double-entry accounting which entries should be positive or negative, but if you remember the "must add to zero" rule you'll be more likely to get it right. Money flowing
into an account is positive, money flowing
out of an account is negative. For credit cards, the money flows "out of" your credit card and into the restaurant's ownership, so the sign should be negative. When you pay the credit card bill later, the sign will be positive on the credit card account (and negative on your checking account, thus again adding to zero) because money is flowing
out of your checking account and
into your credit card account.
Now, look at that double-entry accounting. When you look at the Wednesday Jan 5th entry, and you see that Alice paid you back $30, you'll start searching for a $30 transaction earlier, and you'll pretty quickly find the January 1st and figure out that she owed you $30 because you had paid her share of the restaurant bill on the 1st. And even if the amounts don't line up (let's say she paid you $20 on Jan 5th and $10 on Jan 12th), there's still a "money Alice owes me" category which has a +$30 entry on the 1st, then -$20 on the 5th and -$10 on the 12th, all of which makes it pretty easy to figure out what Alice is paying you back for.
So by recording each entry in at least two places (it's not always exactly two places, e.g. the January 1st expense is recorded in four places total), you get more linkage between the items and it becomes a lot easier to see why the money was going out or coming in.