The fly-half returns to Glasgow on a permanent contract after impressing during a short loan spell earlier this year.
Brandon played three times for Glasgow and scored a try on his debut against Ulster.
He can play at fly-half and full-back and has scored 26 points for the Stormers in seven Super Rugby appearances.
He has also represented South Africa U20 at the Junior World Rugby Championship and has played for Western Province in the Provincial Cup and Currie Cup.
His current Stormers teammates Huw Jones and Oli Kebble will also join this summer and other additions to the squad include Adam Hastings from Bath, Callum Gibbins from the Hurricanes, Lelia Masaga from the Chiefs and Samuela Vunisa from Saracens.
Speaking to glasgowwarriors.org, Brandon said: "I'm very excited to be heading back to Scotstoun. I really enjoyed my short stint there earlier in the year and I’m looking forward to many more good times.
“The relationships I built in Glasgow was definitely a big contributing factor to going back, they are a great bunch of guys and made me feel at home right from the start.
"Dave Rennie is an excellent coach and has done very well with the Chiefs, so I'm super excited to have the opportunity to learn and be part of his set-up.
"The likes of Huw (Jones) and Oli (Kebble) also being there is awesome, it's like bringing a little part of home with me, and I'm sure we'll do great together in Glasgow colours.
"The crowds at Scotstoun are absolutely amazing. To have a packed stadium week in, week out really makes playing there so exciting. I can't wait to play in front of the Warriors supporters again."
Glasgow Warriors assistant coach Jason O'Halloran added: "Brandon really impressed during his short loan spell earlier this year and we're delighted he has chosen to join us on a two-year contract.
"He is able to play in several positions across the backline and will add to the competition we have at fly-half with Finn (Russell), Peter (Horne) and Adam (Hastings) all battling for the number 10 shirt."