Why do some right-handed people bat left-handed in cricket and baseball?

 For cricket, it depends on two main things -

1. Top hand grip or bottom hand grip
2. Left-footed or right-footed.

The lesser likelihood of getting LBW, destroying bowlers’ lines are all additional unintended gifts.

1. Grip: A batsman chooses top-hand or bottom-hand grip based on his comfort or convenience. So, if you are right-handed and a bottom-hand player primarily playing square and cutting, you choose right-hand batting whereas if you are left-handed and more of front-foot driving, you choose to bat left-handed.
2. Front Foot is a major factor in one’s batsmanship - it’s the pivot for all your shots. I am a left-footed. So, I could move my left leg more comfortably and instinctively than my right. With an RHB stance, my left would be the Front Foot and thereby helps me to move freely. When I play with LHB stance, my front (right) foot would just stuck there. Similarly, a right-handed batsman may choose a LHB stance, if he’s more comfortable in moving his right foot as a pivot.
   My theory is right-handed LHB batsman like Ganguly and Lara prefer to play front foot more and could hit sixes easily for this reason.

Read this interesting article “On the Other Hand”[1] on why and how Hussey, a right-handed shifted to batting with LHB stance.

Hussey is naturally right-handed. He writes right-handed, plays tennis right-handed, brushes his teeth right-handed, picks up a spoon right-handed, and throws and bowls with his right arm. When he first picked up a cricket bat, he picked it up right-handed. But on that fateful sunny morning he decided to try batting left-handed, like Border, and ended up sticking with it for the rest of his life.

In so doing, Hussey may well have inadvertently bequeathed himself a natural technical advantage, for if there is one thing that the two main schools of batsmanship that exist in Australia - the Marylebone Cricket Club (MCC) school and the native autodidactic school headed by Sir Donald Bradman and Greg Chappell - agree on, it is this: a grip with a firm top hand and loose bottom hand is optimal for good batsmanship.

Logically it is easier for a batsman who holds the bat with his naturally stronger hand as the top hand (and his naturally weaker hand as the bottom hand) to grip it with a firm top hand and a loose bottom hand. When Hussey switched to batting left-handed, his naturally stronger right hand became his top hand. That wasn't what motivated his change - he did it "purely" because he "wanted to be like Allan Border" - and even when he became a world-class batsman, Hussey was generally not conscious of "the dominance of one hand over another", except when batting at the death of a one-day or T20 game. It was then, he told the Cricket Monthly, that he took the firm top-hand, loose bottom-hand grip to its logical apotheosis:

"At the end of a one-day game or a T20 game, when you're looking to basically hit sixes every ball… I made a conscious effort to really loosen the grip of my bottom hand. So I'd basically just rest the bottom hand [on the bat] on one finger - my index finger - because I was finding that when I was looking to slog, even though my bottom left hand was my less [naturally] dominant hand, it was gripping the bat too hard and taking control of the bat too quickly and affecting my swing. I wasn't hitting through the line of the ball as well as I would have liked."

Footnotes

[1] On the other hand