Taking out your equity when refinancing means that you take out a new loan for the full value of your house (perhaps less 20% as a down payment on the new mortgage, otherwise you'll be paying insurance), pay off your old lender, and keep the rest for yourself.
The result is much the same as using as a HELOC or home equity loan (or a second mortgage), except it's all rolled into a single new mortgage. The advantage is that the interest rate on a first mortgage is going to be lower than on HELOC or equivalent, and the equity requirements may be lower (e.g. a HELOC may only let you borrow against the amount of equity that exceeds 25% or 30%, while a new mortgage will require you only to have 20% equity).
This is especially attractive to those whose homes have appreciated significantly since they bought them, especially if they have a lot of high-interest debt (e.g. credit cards) they want to pay off. Of course, rolling credit card debt into a 30-year mortgage isn't actually paying it off, but the monthly payments will be a lot lower, and if you're lucky and your home appreciates further, you can pay it off fully when you sell the property and still have paid a lot less interest. The downside is that you have turned unsecured debt into secured debt, which puts your home at risk if you find yourself unable to pay.
In your case, you don't yet have even 20% equity in your home, so I wouldn't recommend this. :-)
Equity is simply the difference between the amount you still owe on your home and the amount you'd get if you were to sell it. Until you do sell it, this amount is tentative, based on the original purchase price and, perhaps, an intervening appraisal that shows that the property has appreciated. That is really all that it is and there's nothing magic about it, except that since you own your home, you have equity in it, while as a renter, you would not.
It used to be (decades ago, when you needed 20% down to get a mortgage) that selling was the only time you'd be able to do anything with the equity in your home. Now you can "take it out" as described above (or borrow against it) thanks to various financial products.
It is sometimes tempting to consider equity roughly equivalent to "profit." But some of it is your own money, contributed through the down payment, your monthly principal payment, and improvements you have made -- so "cashing out" isn't all profit, it's partly just you getting your own money back. And there are many additional expenses involved in owning a home, such as interest, property taxes, maintenance, utilities, and various fees, not to mention the commissions when you buy or sell, which the equity calculation doesn't consider.
Increasing equity reflects that you own a desirable property in a desirable location, that you have maintained and maybe even improved it, that you are financially responsible (i.e., paying your mortgage, taxes, etc.), and that your financial interests are aligned with your neighbors. All those things feel pretty good, and they should. Otherwise, it is just a number that the banks will sometimes let you borrow against. :-)