How can I explain why it is important to make new friends?

There are some good answers, but I wanted to speak up a little for us introverts to provide some counterpoint.

There's a big difference between being able to get along with new people and liking to meet new people. As hard as it is for extroverts to believe, introverts are more comfortable alone. When we want to unwind, we seek solitude rather than a night on the town. That doesn't mean we don't have a small group of close friends, or never want to be social, just that it expends mental energy for us to be around others rather than being rejuvenating.

It's not healthy to take to extremes, but everyone has a different balance point. If an extrovert has 5 free evenings, maybe he will want to spend 4 with other people and one at home by himself. An introvert may have the reverse preference, but they both have a good balance. By all means, encourage a child who never has any social interactions to do so, but don't worry if he goes a couple weeks in between, or prefers one-on-one or very small gatherings. You don't have to enjoy working a crowd to be happy and have friends.

1. Fun: Not all people are fun. But the more people you meet, the more fun people you will meet, and the more fun you will have, and it will be easier to avoid the people who are not fun.

2. Growing: Great people usually comes in groups. I once saw a documentary about British punk. They talked about a group of kids (several from the from the same suburb in London) that used to go into London to see early Sex Pistols concerts. That group of friends included the people like Siouxsie Sioux, Adam Ant, Steven Severin, Billy Idol (yes, they all took silly names) who all became respected and well known rock stars. While at Santa Monica High School, Charlie Sheen, Emilio Estevez, Rob Lowe and Sean Penn made movies together. Is it chance? No. Groups like this inspire and push each other to get better. They don't just recognize talent in each other, they create talent in each other. It doesn't matter if it is music, movies, computers or science, and it can be in high school, university or societies. If you want to be great, you have to find like-minded people and be greater together.

3. Career: The last job I got without using my contacts was in 1991. If you can't make friends, nobody will recommend you to their friends, and you'll get stuck in crap jobs.

4. Help: When you have a tricky problem you need help with, maybe it's studying in school maybe you need someone to help you when you are repairing your bike, or your car, or to help you carry the sofa when you move, the more people you are friends with, the more people will help you. This of course means you will help them too, and everyone gets help. That's what friends are for.

Making friends is important because:

1. The world is full of people, so there's no chance that you never have to interact with anyone.
2. Things are easier to accomplish if others help you because they can help you solve problems and think up new ideas.
3. It's more fun to do things together because you have someone you can share the joys and triumphs with.
4. Sometimes you meet people that are hard to get along with. Sometimes you will have to deal with them anyway. Learning "people skills" helps in that situation. You develop "people skills" by being with people.
5. If you're among friends (=people who like you) and you make mistakes, then they will be kinder to you than strangers would. When you learn to avoid such mistakes, strangers are nicer to you.
6. Sometimes you meet people who want you to do things you don't want to (games, assignments, etc.). It takes practice to avoid those things, or to work toward a solution everybody is happy with.
7. When you're older and have a job, all of the above is even more important because you can't just (leave the playground/go home) but you have to deal with the other people.

Well, I like to play Rock Band.

And I can tell you that Rock Band is a lot more fun when I play it with some friends than when I play it alone.

I'm sure the same basic advice goes for many kinds of games -- board games, card games, hide and go seek.. etcetera. It's just more fun to play with friends, isn't it?