Charts are generally read right to left on RS and left to right on WS (if all rows are charted). If the chart is numbered, as this one is, you can see if every row is charted or just RS ones (if WS rows are not charted, the pattern usually tells you to purl (or whatever) every WS row). The RS rows are the odd numbers on the right, the WS rows are the even numbered ones on the left. You must follow the legend (or key) as to what each st is on which side - on this chart, a blank box is a knit st if you're on the RS, but it's a purl if you're on the WS; a dot is purl on RS, a knit on WS.
So on this pattern, after you cast on and knit 3 rows, you'll start on the chart. The first and last 3 st of every row are knit as a border, and are not included in the chart. You'll knit your 3 border st, then start on the chart in the lower right corner, where it says "1" and work across to the left, finishing that row. Turn your work and do a purl row (all blank boxes for row 2). Since the first 11 rows of the chart are not patterned, it's just a knit on RS and purl on WS. On row 12 (WS row), you'll start your pattern st, reading from the left side of the chart where it is marked row 12 - so knit the 3 border st, then PURL for 15 st (the 15 blank boxes on the chart), knit 1 (the dot), purl 15 more. You have completed the chart 1x. It wants you to do that 2 more times, so continue purling 15 more st, knit 1, purl 15 (2x completing chart), then repeat for the 3rd time and then knit your 3 st border. Turn your work.
You are now on the RS of your work and reading right to left where it is marked row 13. So, knit your 3 border st, knit 14 (blank boxes are knits on RS), purl 3 (the dots, now purls since you're on RS), knit 14 more, repeat * to * 2x more for the 3 chart repeats it wants, knit your 3 border, turn you work and start row 14, continuing to read the pattern as described.
I know it sounds really complicated, but it's not once you get into the swing. You may want to tie a piece of waste yarn or attach a coilless pin to the RS of your work so you can tell at a glance where you're at - not so critical with this pattern since it's a fair amt of knit st visible on your RS, but on other patterns it may be difficult to tell which is which until a large portion of the pattern emerges.