COUNTIF simply counts the number of times each value appears in the data range. By definition, each value must appear at least once, so when the count equals 1, the value is unique. When the count is 1, the formula returns TRUE and triggers the rule.
Conditional formatting is evaluated for each cell that is applied to. When you use a formula to apply conditional formatting, the formula is evaluated relative to the active cell in the selection at the time the rule is created. In this case, the range we are using in COUNTIF is locked with an absolute address, but A1 is fully relative. So, the rule is evaluated for each of the 40 cells in A1:D10, and A1 will be updated to a new address 40 times (once per cell) while $A$1:$D$10 remains unchanged.
Named ranges for a cleaner syntax
Another way to lock references is is to use named ranges, since named ranges are automatically absolute. For example, if you name the range A1:D10 "data", you can rewrite the rule with a cleaner syntax like so: