Welcome to KH!
Generally, if your pattern is available online it's helpful to include a link to the pattern .... people can usually provide more help if they can see the pattern themselves. If there is no link, you can always do what you did and just type out the part of the pattern that is confusing you (just be sure not to post the whole pattern, since that would be a copyright infringement).
From what you have described, it sounds fairly straighforward though. It sounds to me like you would want to do the following:
1) Continue working in garter stitch. I assume this is what you have been doing so far. In case you're not familiar with the names of different stitches yet, garter stitch is created by knitting all rows (ie. you're not doing any purl stitches at this point).
2) You want to increase (the abbreviation "inc") one stitch at each end of your next row. There are different ways to increase - sometimes your pattern will tell you what method they want you to use (be sure to read your whole pattern, including any abbreviations listed at the beginning of the pattern to see if you are supposed to use a specific increase) and sometimes it's up to you to use what you want. I would definitely recommend checking out the "Videos" page here on KH, as there is a whole section on different increases. In this case, you would probably be fine to use either an M1 ("make 1") or a KFB ("knit front and back"). Whichever you choose, you would do this once at the beginning of the row and then once at the end of the row.
3) Then you want to increase one stitch at each end of the "6 following 6th rows". I take this to mean that you want to increase every 6 rows 6 times. So after you finish your first increase row (step 2 above), you would knit 5 rows even (meaning you don't do any increasing) and then on the 6th row you follow step 2 above again to do another increase row. You keep doing this (knit 5 rows then increase on the 6th row) until you've done a total of 7 increase rows (your first increase row plus the 6 times you repeated it afterwards).
Hopefully that helps you get started, but if not just let us know what's still confusing you!
I disagree with Trvvn's interpretation, but maybe I'm misunderstanding.