LTE-LAA is a version of LTE designed to work in radio frequency bands (spectrum) that are designated by the government as unlicensed, such as the 5 GHz band, which is also used for Wi-Fi.
LAA is like standard LTE, but with the added ability to detect other technologies (like Wi-Fi) operating in the same unlicensed band in that area, and coexist without interfering.
Traditional LTE (like most cellular technologies) operates in licensed bands. The government sells licenses for specific radio bands to mobile network companies (carriers), when then operate mobile networks in those bands. No other radios are allowed to operate in those bands.
But there are also bands that are designated as unlicensed, which are commonly used for technologies like Wi-Fi and Bluetooth. Because these are unlicensed bands, you do not need a license from the government to set up your own Wi-Fi network.
Technologies designed for unlicensed bands are designed such that multiple networks can be set up in the same area, using the same band, without directly interfering. (However the band can still become congested if too many people are transferring too much data using one or more networks in the same area.)
LAA works in conjunction with carrier aggregation, which allows a phone to use multiple bands at once to boost data speeds. Carrier aggregation is a required part of LAA. When a mobile device is using LAA, it must also be connected to a traditional LTE network operating in a licensed band. Therefore LAA cannot be used on its own for connectivity; it can only be used to enhance an existing LTE network (operating in licensed bands), improving capacity and data speeds.
See: Carrier Aggregation
LTE-U is a very similar and related technology.