Why should I use MailChimp or similar instead of a custom made script?<>
This question already has an answer here:
Strictly speaking there is no reason you can't do what Mailchimp (and others) are offering. However they do tend to make it a lot easier to work with.
They all have their different approaches, but they tend to speed up development of complicated mail campaigns and allow them to be managed better. This is particularly true of HTML emails, which can be an absolute time sink to get right on all browser & mail client combinations. (Gmail on IE8-11, Mail app on iOs, Outlook on Mac... etc. etc.)
I'm speaking specifically of Mailchimp here, but this should apply to many others:
Templates. They provide HTML email templates to get your designs started. This can avoid a lot of subtle errors in HTML mails.
Device preview. They provide a preview service to display your mail on a large number of devices & mail clients. So you don't have to go through them all yourself.
Opt out service built in. In Europe at least all marketing emails must have a link to unsubscribe from the mailing list. It's a legal requirement that can bite you and they handle this neatly. The link is always embedded and you can't accidentally re-add someone when you create another list.
Scale. They are built to sent lots and lots of mail. This can reduce the load on your servers and keep them from falling over. It can be particularly useful to make sure mail gets delivered quickly and doesn't sit around in a mail-queue.
Management. You can set up the campaign (as a developer) and hand over to the marketer(/someone else) to manage the campaign. => More time on SO for you!
Spam. They may come with an anti spam tool to check your mail before it's sent out. Not foolproof, but handy before you spend your money on a campaign.
This may sound like an endorsement for Mailchimp, but any of these kinds of services will do much the same. There are many of them so shop around and find the best fit for you. My main point here is don't re-invent the wheel. Emails may seem simple, but can get complicated fast.