The intermediate carry flag, or "adjust flag", or half-carry flag is used to facilitate binary-coded decimal (BCD) arithmetic, where each decimal digit of a number is represented as a nibble (a group of 4 bits). The range of valid values for each nibble is 0 to 9 (0000 to 1001). If, after an arithmetic operation, the result contains a "non-decimal" nibble (1010 to 1111), it can be deduced from its bit pattern that a correction is necessary. However, it might also be necessary even if both nibbles of the result look "decimal".
Here's an example (from Wikipedia) why the flag is needed:
Adding the BCD values 3916 and 4816 produces 8116. This result does not have a non-decimal low nibble, but it does cause a carry out of the least significant digit (lower four bits) into the most significant digit (upper four bits). This is indicated by the CPU setting the half-carry flag. This value must also be corrected, by adding 0616 to 8116 to produce a corrected BCD result of 8716.
0011 1001 39 + 0100 1000 48 ----------- 1000 0001 81, intermediate result + 0110 06, adjustment ----------- 1000 0111 87, adjusted result