Assets = Liabilities + Equity
What you own = what you owe + what is actually yours
Assets + Expenses = Liabilities + Equity + Income
everything a debit increases
everything a credit increases
Income and expenses are equity in motion
Earning revenue makes the business worth more to its owners; spending makes it worth less. Rather than adjust equity on every transaction, the books keep income and expense accounts for the year and roll the difference — the profit — into equity at the end of it. That is the whole reason for the expanded form above.
Balanced by construction
Put the same total on both sides of an entry and the equation cannot break, whatever the entry was. It is not that bookkeepers are careful; it is that an unbalanced entry is not a valid entry, in the same way a sentence with no verb is not a sentence.