First this "Matter can neither be created nor destroyed".
This idea is not the entire picture, it is more correct to say that matter and energy are conserved as a whole, but you can convert between the two.
As far as dating things, you're not dating the matter, you're dating the state in which the matter is. For instance, if you want to date a bone, you're not interested in how old atoms making up the bone are.
Most dating methods are based on unstable isotopes, isotopes such as Carbon-14, Uranium-238 have too many neutrons to maintain a stable atom, so eventually some form of radiation is given off, either alpha radiation, a particle consisting of 2 protons and 2 neutrons, beta radiation, also a particle, this time a single electron given off as one neutron decays into a proton and electron, and gamma radiation, a highly energetic photon, which usually is released along with one of the other forms.
Each time an atom loses mass by emitting a particle it changes it's composition, for instance U-238 emits an alpha particle, and converts to Thorium-234. This is the basis of dating, knowing what percentage of various elements in an object.
An example, you buy a brand new block of U-238, you know there is nothing that block but U-238. Yu put the block in a container and leave it for a few years, when you get it back out, you bust out your Ronco Home Mass Spectrometer and analyze some samples, you find that now it's made up of Uranium, Thorium, Lead, etc. If you knew how long it took for a given amount of Uranium to turn into the other elements, then you could use your data to determine how long the uranium block sat there. Dating methods work on the same principle, I take a rock, I analyze the elements in it, in almost everything there is at least one, and usually a few elements that can be used for dating. Since we can measure the time for the elements to decay, we can extrapolate backwards to get an estimate of how long the object has existed in it's current form.
Carbon-14 is common dating method for things that were previously alive, but have since died. Living organisms use carbon as a fundamental element, and there is a certain percentage of carbon in a living creature that is C-14. This percentage is more or less constant as long as the object is alive, but once it dies it no longer is renewing the carbon, and the C-14 begins to decay. Since we have very good data on howw much C-14 is in current living things, and we can make assumptions that the levels were similiar in the past, we can simply analyze an object to determine that C-14 content, and extrapolate from that how long ago the thing died.