Recursion is much easier to understand if you use goto instead of functions. Functions are a high level concept in the C language (and most other languages) but it gets compiled down to (essentially) the older goto style of programming which is much easier to understand.

goto 42 will move execution to line 42 of the code. goto x will move to the line of code labeled x.

Most modern languages don't even have goto support, since functions are cleaner and tend to produce more easily maintained code, however as a programmer you should be aware what's going on under the hood. Here's your code rewritten to use goto:

int main(void)
{
    int height = get_int("Height: ");
    int row = 1; // Starting row
    int col = 0; // Starting column

draw:
    if (row > height) // Exit condition
    {
        goto end;
    }

    if (col < row)
    {
        printf("#");
        col++;
        goto draw; // Repeat the same row
    }
    
    // Move to the next row
    printf("\n");
    row++;
    col = 0; // Reset column for the next row
    goto draw;

end:
    return 0;
}

PS: goto is also how for loops, if statements, switch blocks and others work under the hood.