1. You don't feel obligated to visit every tourist attraction.
You don't need to climb the Eiffel Tower twice, unless you're traveling with friends who have never been to Paris. Not only does traveling alone save you the time of doing things twice, but you don't have to spend €40 a day doing things you've already done.
2. Lodging is cheaper for one person.
Whether you sublet a room in an apartment or find a room on Couchsurfing.
3. You're more likely to save money on food by skipping restaurants.
I eat candy bars for breakfast more often than I would like to admit. Hey, it's cheaper than spending €15 every morning at a cafe!
4. People give you things for free when you're alone.
Case in point: I went to a museum yesterday, and since I was the only one in the lobby, the woman behind the register just waved me through.
5. You can create a budget based on your needs, rather than the needs of a group.
I would rather spend money on historical sites than eating at a restaurant twice a day.
6. You're more likely to meet locals when you travel by yourself.
Locals show you inexpensive deals that you can't find on a basic internet search. They're also more likely to invite you to have dinner at their house, spend the night on their couch, and go to their friends' birthday parties.
7. You're more likely to sacrifice your comfort to save your budget.
Did I mention candy bars for breakfast? This applies to everything, from second class seats on a train to cheap food.