Interest rates affect currency because when Japan's central bank raises rates, it becomes more attractive to hold yen - you get better returns on savings and investments denominated in yen. So investors globally start buying yen to take advantage of those returns, which drives the value up. It's pretty straightforward supply and demand.
For most regular people, the impact depends on what you actually do. If you're importing goods from Japan or planning a trip there, you'll feel it in your wallet immediately - everything costs more when the yen strengthens. If you're exporting to Japan or have investments there, the opposite happens and you benefit. Most folks though won't notice much in their day-to-day life unless they're specifically dealing with Japan.
Here's the thing people often miss though: currency movements can be surprisingly volatile, and the yen in particular has a reputation for wild swings. So if you're thinking "oh, rates are going up so yen will stay strong forever," that's not necessarily true. Markets overshoot all the time. The yen could just as easily weaken again if economic conditions change or if the Bank of Japan signals they're done raising rates. Don't make any major decisions (like booking a trip or making purchases) assuming today's exchange rate will stick around for months.