You're not reimbursed for those particular items in any way. You just get to deduct the cost from your income. For example, if you pay $100 for a piece of software, you must have gotten that $100 as income somehow. Roughly speaking, the tax writeoff means you get to pretend you never earned that $100 as income in the first place.
How much money you save from this depends on your marginal tax rate. The higher your tax rate, the more you would effectively save. For instance, if you were in the 25% tax bracket, then earning an additional $100 would mean you must pay $25 in taxes. So if you get to deduct that $100 and not count it as income, you effectively save $25 by not having to pay tax on the $100.
You "save" this money only in the sense that you don't have to pay it in your tax bill when you otherwise would. No one writes you a check reimbursing you for the $100 you spent. It just means your tax bill will be slightly smaller, or your refund slightly larger.
This is the basic idea, but I'm glossing over a lot of complications here. There are various restrictions on when you can or can't deduct a certain expense, and in some cases you can only deduct a portion of the expense, so paying $100 on a work expense may not actually reduce your taxable income by $100. But the idea is the same: your effective savings are a percentage of what you get to deduct from your income, and that percentage is larger the higher your marginal tax rate.
