Sendgrid uses a mail transfer agent like Postfix or PowerMTA (I don't actually know what they use but I wouldn't be surprised if it was PowerMTA) to talk to receiving email servers via SMTP.

It's not hard to set up the technical bits to send email. You basically just need to run an MTA that knows how to stamp your outgoing messages with DKIM (a cryptographic signature). Your server has to have an SSL certificate. You also need SPF records set up that accurately reflect what servers are allowed to send on your behalf. DMARC records are helpful as well but not required. If you're only sending as yourself, that's basically it. You should be able to get mail through to the big receivers without much trouble (maybe having to mark "not spam" a few times to start).

The trick to maintaining your sending reputation as an individual is to not send email for anyone else and never send content that you don't control. Sendgrid sends for other people and they have people on staff who's full time job is to fight with the big receivers to maintain their reputation.

Sendgrid puts a huge amount of money and resources into maintaining both technical and political legitimacy. They have relationships with other major senders and receivers of email, they follow every single rule to the T, they aggressively police their own IPs (of which they have many) etc. It's a complicated business, which is why it's totally worth it to pay them to do it for you.