Do a Google search for Jarawa and you'll see that these are people clearly of African origin. Did they end up on islands under Indian sovereignty by continental drift or did they sail eastward and settle on the Andaman Islands?
This is really a question for anthropologists, not historians. You might have better luck posting to /r/askanthropology.
Briefly, no it has nothing to do with continental drift, which happens on geological timescales, not human timescales. In all the time the human species has existed, the continents haven't moved much - maybe a couple miles or so.
The Andaman Islanders arrived by boat. Their origin is an area of active research in genetics, and the question is not resolved. Here are a few things we do know, though:
Despite the fact that they look African, as you say, they are definitely Asian in their genetics. They belong to mitochondrial haplogroup M, which is prevalent in south and Southeast Asia, not haplogroup L which would indicate recent African origin.
However, their particular subgroup of M is different from other M subgroups in Asia, indicating that they have been isolated from other Asian populations for a very long time.
From genetic dating, it looks like the separation is in the range of 30,000 - 60,000 years, but genetic dating is very inaccurate given the tiny sample sizes. We can say with more certainty that they avoided mixing with the huge migrations of farmers that happened about 8,000 years ago, so their isolation is definitely longer than that.
The Jarawa have no Denisovan admixture, which points towards an earlier rather than later date for their isolation.
So it sort of boils down to that they are the descendants of one of the earlier migrations out of Africa, probably 60,000 years ago, which followed the southern coastal route across India and Southeast Asia. They were mainlanders before they became islanders. They traveled to the islands on boats. When they did that, we don't know, but the genetic evidence points to somewhere between 8,000 - 30,000 years ago. The earlier date is more likely.
As for their resemblance to certain African populations, it may simply be adaptation to a similar tropical forest lifestyle. We do not know what the ancestors of these islanders looked like 30,000 years ago. They may have looked less African than they do today.
The indigenous Andamanese are what are known as 'negritos'. Similar populations are found in remote parts of Malaysia and the Philippines.
Essentially, most of South-East Asia was historically populated by negritos, until later groups of people migrated southwards, possibly from Taiwan or Southern China, who supplanted the original negritos.