"A tad" is a more informal way of saying "a little bit". Similar to saying "a pinch", or "a smidge". Example: "I'll be running a tad late today." "It's a tad chilly outside."

"A tad closer" simply means a little bit closer. An example could be: "The other runner ran a tad faster than I did, and beat me to the finish line by a few seconds."

There's an expression that goes like "Tit for tat" which means to counteract USA showed they have 70 nuclear bombs so Russia showed that they have 80, it was a tit for tat kind of situation so when you hear the word tad closer I think it comes from the expression Tit for tat. To equalize / counteract / become closer