The standard algorithm in the US (though this might not be the standard everywhere) is to add the digits in the ones place (carrying the 1 - which represents 10 - if their sum is 10 or more), then to move to the tens place and add the digits there (again carrying the one if their sum is more than 10 - though in this case the one that represents 100) and then we move to the hundreds place and so on.
So, let's do this with the numbers given. 57+58. We would line these up one over the other so that the digits 8 and 7 which are in the ones place are above each other. We add these and get 15 which is one ten and five ones. The 5 gets put down and the 1 gets carried over. Now we focus on the digits in the tens place (here 5 and 5). We add these and to their sum add the 1 we carried. This gives us 11 (but in actuality it means 110). We bring down the leftmost 1 and carry the other. There are no digits in the hundreds place so we being down the 1 we carried earlier.
This gives us an answer of 115.