"My internet connection has become terribly slow due to a trojan/spyware which is eating up my internet bandwidth."
What process/software did you use to determine this?
"However, apparently there is no process consuming bandwidth according to the monitor program I am running."
Then how do you know this has anything to do with malware?
"How is it? How do trojans and spywares manage to eat up bandwidth?"
Software that consumes bandwidth (malware or not) uses bandwidth by sending and receiving data, exactly what your bandwidth is for. What data is being sent, where it's going to, and what is being done on your computer locally, are all questions that when answered, will answer your question.
Your question is the subject of many good books, and as such, may not fit the Q&A format of Superuser. There isn't enough space/time for me to go into every piece of malware and explain its use of bandwidth (if in fact the problem you are experiencing is due to malware of some sort).