I cannot explain how a TDS meter works, as I have never used one. I may be able to help with some explanation about TDS and fish (taken from an article I wrote a few years back). This is linked to what I just wrote in response to your other thread on pH.
TDS (total dissolved solids) is basically everything dissolved in the water: chlorine, chloramine, ammonia, phosphate, salt, hard minerals (GH), bicarbonates (KH), organics, etc. And almost every substance added to the water will increase TDS: water conditioner, fish foods, plant fertilizers, calcareous substances, medications, water adjustment products, etc.
Fresh water by definition contains no more than 1500 mg/l of TDS. Brackish water contains 1500-5000 mg/l, and marine (salt) water has more than 5000 mg/l of TDS. Note that mg/l is basically equal to parts per million (ppm), and also that this is not suggesting a level of 1500 ppm in an aquarium; these are just the approximate figures for the three categories.
TDS impact fish. Hard water fish--as we term those species such as livebearers, rift lake cichlids, and some of the atherinids, cyprinids and catfish—can withstand higher TDS than soft water fish. The TDS in Lake Tanganyika is around 400 ppm. Compare this to the near-zero TDS in many Amazonian streams.
Fish live in water, and their bodies contain water; the fish’s cells separate these two waters, but the cells are semi-permeable, which means the cell will permit the movement of water and certain non-polar molecules to pass through either way (called osmosis) but will prevent the passage of larger or charged molecules. The way the water moves is determined by the difference in concentrations between the two waters: water of higher concentration (more dense) will attempt to pass through to the water of lower concentration (less dense) until the two are equal. If the fish could not somehow control this natural flow, it would either rapidly dehydrate or explode. But fish are able to control this through osmoregulation, a complex series of chemical processes. The water moving in or out of the fish’s body will likely have a different pH, so another set of processes controls the function of regulating the pH of the fish’s blood (Muha, 2005). Both of these processes also affect the ability of the blood to carry oxygen, and this impacts many other functions including digestion, the immune system, and so on.
TDS accumulate obviously, so the more water changed regularly, the better. Another reason for substantial weekly water changes.
You specifically asked about Betta; being a soft water fish, the TDS should be as low as possible. The GH has the greatest impact here, so having soft water is a step in the right direction. Then. limiting additives to those absolutely essential. We cannot not use conditioners and risk poisoning the fish with chlorine/chloramine and perhaps heavy metals, but we should attempt to keep TDS low. This is why it is foolish to use more conditioner than needed; manufacturers will still say to dose for the tank volume when changing say 50% of the water, but this is doubling the TDS and affecting the fish, regardless of how "safe" they try to suggest this is for fish. Common sense is a better safeguard.
Byron.