Suppose $0.
Suppose the probability distribution of $X$ is a memoryless distribution on the interval $(0,\infty)$.
Commonplace mistake: $\Pr(X>b\mid X>a) = \Pr(X>b)$.
Correct statement: $\Pr(X>b\mid X>a) = \Pr(X > b-a)$.
The second thing is what memoryless says. The first thing would be correct if the two events $[X>b]$ and $[X>a]$ were independent. But notice that if $[X>b]$, then necessarily $[X>a]$, so how could they be independent?
So one connection between independence and memorylessness is that memorylessness gets carelessly mistaken for independence in this situation.
But now consider the Poisson process. They waiting time until the next occurrence has a memoryless exponential distribution. This implies that the numbers of occurrences in two time intervals that don't overlap are independent. That's a more substantial connection between memorylessness and independence. Notice what is independent: the number of occurrences in a time interval has a discrete distribution, whose values are in the set $\{0,1,2,3,\ldots\}$. Those numbers of occurrences are what are independent. The things that are memoryless are the distributions of the waiting times. Those have continuous distributions, whose values are in the set $(0,\infty)$.