You need something flat that will hold your board in place. It should be the same height as the PCB. The acrylic jigs available from OSH Stencils work great. Use tape to secure it in place.
Avoid tape anywhere the stencil will touch. You want a rigid, flat surface under the stencil with no extra/bumpy layers.
Carefully place the stencil over your board and tape it in place. You'll know it's lined up when each hole changes to gold.
Again, it's important that everything is flat and that the stencil is touching the PCB as directly as possible. Bowing, even at the edges, will cause trouble.
Apply a generous amount of paste. Then spread it over the stencil's holes using a plastic spreader card.
Try to do this in one pass only! The more you swipe across the stencil, the more paste squeezes under the holes and ends up in places you don't want.
Using a pair of tweezers, place each component from the BOM onto the board. If the labels on the PCB are hard to read, use this printable board layout guide: Board Layout PDF.
The resistors and capacitors can be placed in either orientation.
Use the "pin 1" markings (usually a small white dot or circular depression in the plastic) to orient the ICs. Pin 1 should be in the same corner as the asterisk on the PCB.
U4 is very small, but you can't get it wrong if the "gap" in the pins is on the correct side.
U5 is very small and isn't missing any pins, so you may need to use a loupe or magnifying glass to see the U5 pin 1 stripe. The stripe side should be the one nearest the edge of the PCB.
My U6 placement in the video is a good example of how not to do it. (It's harder with a video camera in the way!)
Heat the board until the solder paste melts.
If you're using low-temp paste, the temperature profile is a lot less critical and you can use just about any heat source (as long as it's not an open flame).
Afterward, check with a loupe or magnifying glass for solder balls. You can usually scrape them away with tweezers.
The bottom of the PCB shouldn't have any exposed copper, so there isn't much risk of shorting anything once installed. That said, it's better to be safe than sorry. Placing a strip of Kapton tape across the back of the mod will provide some extra insurance.
After trimming it at either end of the board with scissors (not shown), your TMS-RGB board should be complete and you're now ready to follow the Installation Guide.