Well, the documentation for BinaryReader.Read7BitEncodedInt already says, that it expects the value to be written with BinaryWriter.Write7BitEncodedInt and that method documentation details the format:
The integer of the value parameter is written out seven bits at a time, starting with the seven least-significant bits. The high bit of a byte indicates whether there are more bytes to be written after this one.
If value will fit in seven bits, it takes only one byte of space. If value will not fit in seven bits, the high bit is set on the first byte and written out. value is then shifted by seven bits and the next byte is written. This process is repeated until the entire integer has been written.
So the integer 1259551277, in binary 1001011000100110011101000101101 will be converted into that 7-bit format as follows:
Remaining integer encoded bytes
1001011000100110011101000101101
100101100010011001110100 00101101
10010110001001100 10101101 01110100
1001011000 10101101 11110100 01001100
100 10101101 11110100 11001100 01011000
0 10101101 11110100 11001100 11011000 00000100
I'm not that confident in my C skills right now to provide a working implementation, though. But it's not very hard to do, based on that description.