Quick and easy way is to think of a foreign currency (let's call this USD) as another good with its own demand and supply.
Imports and exports
When a country imports a lot (assuming it imports from the US or the invoice is in USD), the importers will need to pay for these goods in USD. To do this, they go to a bank (roughly speaking) and buy USD, pay for USD with the local currency. This drives up the demand of USD, and hence the price of USD goes up... the local currency depreciates against the USD.
When a country exports a lot, the reverse happens. Importers in other countries will need to pay local currency, so they sell the USD they have in order to get the local currency. The supply of the USD goes up (or you can think of it as demand of USD goes down)... price of USD goes down, and the local currency appreciates against the USD.
Now to go back to your original question... FX reserve is just a place for a country to hold foreign currencies (since USD can't be used locally). It's like a warehouse for USD, controlled by the central bank.
When a country can export a lot there will be a lot of people who wants to sell USD. The rapid appreciation of the local currency could affect firms (profits, etc.) so the central bank might "intervene" to stabilize the rate in the short run. They do this by buying up USD... in a sense create additional demand for USD so the price of USD doesn't fall down too fast. They then put this USD they bought in the reserve.
This is why the change in FX reserve could be used as a proxy to see how much central bank has intervened in the FX market.
In buying USD, they could print new local currency and buy USD outright, resulting in higher supply of local currency and higher inflation, or they could "sterilize" the action by selling bonds just as they would do when they want to reduce money supply by the same amount that they used to buy USD.
On the other hand, if the local currency is depreciating too fast (price of USD going up), the central bank could help by selling USD in the reserve to increase USD supply.
The problem, though, is that the reserve could run out. This is what happened in the Asian Financial Crisis in 1997.
Hope this helps!