hi annie and welcome to KH!
and also welcome to the wonderful world of DROPS designs… others will say it nicer than me, but some DROPS designs can really suck, as far as directions go. because they’re European and often translated, things can get missed in the translated version. also, they tend to leave out some assumed knitting knowledge that not everybody knows, or that a reminder of would be nice.
in knitting, a garter stitch ‘ridge’ is 2 rows, achieved by just doing knit stitches. cast on, knit, turn work over, knit again. that’s one ridge. that combination makes it look like a knit row followed by a purl row from one side (or a purl row followed by a knit row viewing it from the other side), because you turned the work. (video for it at http://www.knittinghelp.com/video/play/garter-stitch-english)
so in your case, you need to cast on 22 stitches. then knit those 22 stitches (that was your ‘right’ side). then turn your work over. here you need to keep knitting those 22, but also increase by 6 stitches. if the directions don’t tell you exactly where to increase, a good guess is to increase evenly. for 22 i’d knit 4, then increase 1, then knit 3 then repeat between the **s that will give you 28 stitches total. (this was your ‘wrong’ side). it doesn’t say how to increase, unless it’s in a different part of the instructions. if you know an increase style you like, use it… knit front&back, make 1, make 1 left, make 1 right, are all good options. (videos at http://www.knittinghelp.com/videos/increases show various styles too)
next you’ll follow the cable pattern. which they didn’t label fully, but unless otherwise stated/marked, usually a table like that is read from the BOTTOM RIGHT corner, working to the left, the way you would actually knit. then row 2 working to the right, row 3 working to the left, etc etc. so here it should be:
Row 1 (Right Side): K5, P2, K6, P2, K6, P2, K5
Row 2 (Wrong Side): K3, P2, K2, P6, K2, P6, K2, P2, K3
Row 3 (Right Side): K5, P2, Cable (Slip 3 front-wise to holder, K3, Slip 3 back onto needle & knit), P2, Reverse Cable (Slip 3 backwards to holder, K3, slip 3 back onto needle & knit) , P2, K5
Row 4 (Wrong Side): Repeat Row 2 - K3, P2, K2, P6, K2, P6, K2, P2, K3
Row 5 (Right Side): Repeat Row 1 - K5, P2, K6, P2, K6, P2, K5
Row 6 (Wrong Side): Repeat Row 2 again - K3, P2, K2, P6, K2, P6, K2, P2, K3
Row 7 (Right Side): Repeat Row 1 - K5, P2, K6, P2, K6, P2, K5
Row 8 (Wrong Side): Repeat Row 2 yet again - K3, P2, K2, P6, K2, P6, K2, P2, K3
you’ll work that 8-row repeat until your item is the length you want, somewhere between 32-35 cm (12.5-13.75") in the pattern, but will depend on your desired fit. the cable needle only has stitches on it when you’re working those 3 stitches at a time in row 3 of the pattern. otherwise it just sits on your work surface ready to go. it doesn’t stay attached/knitted into the work the whole time. you use it to hold stitches, then move them off the cable holder to knit using regular needles. then it’s empty, ready for when you get to the next cabling instructions. (video at http://www.knittinghelp.com/video/play/crossing-cables-with-a-cable-needle)
once your whole thing is the right length for you, the directions say to work another ridge in garter (knit the whole row, turn work, knit the whole row again) and then bind off. then you’ll do the button decoration if desired, and finally sew the cast on and bind off edges together to form a seamed piece.
hope that helps, and if you need anything else there’s lots of knowledgeable people here