Why in Overlap Save Method in DSP output length comes out to be same as input length? I  know that when we do convolution of $x(n)$ (Length $L$) and $h(m)$ (Length=$M$), length of output data comes out to be $L+M-1$, but when we break a large sequence of data and find output using Overlap Save Method, length comes out to be same as input.

Why do the extra $M-1$ points are appearing in this case?
In more expressive way, be $y(n) = x(n)*h(n)$:


Suppose Case 1.:

$x(n)$ (length $L$),
$h(n)$ (length $M$), leads to
$y(n)$ (length $L+M-1$)

using simple linear  convolution
Case 2.:

$x(n)$ (length $3L$),
$h(n)$ (length $M$), leads to
$y(n)$ (length $3L$)

using the Overlap Save Method that evaluats output after breaking input in 3 equal parts of
length $L$.
Your "Case 2" convolution using overlap method processing really comes out to 3L + M - 1 with M - 1 (or more) points discarded.   And another 2(M - 1) points computed and summed into the middle of those 3L points that you see.

The key concept is in the word "overlap".

So no extra points are appearing because they're always somewhere (summed or trashed).