One of the apps you have a picture of, the venerable GPS Status & Toolbox, describes this claim in the app description:
Keep your GPS fast: reset it or download A-GPS data regularly for faster fixes
Wikipedia does a great job defining the basic concepts behind Assisted-GPS:
Standalone GPS provides first position in approximately 30–40 seconds. A Standalone GPS system needs orbital information of the satellites to calculate the current position. The data rate of the satellite signal is only 50 bit/s, so downloading orbital information like ephemeris and almanac directly from satellites typically takes a long time, and if the satellite signals are lost during the acquisition of this information, it is discarded and the standalone system has to start from scratch. In AGPS, the Network Operator deploys an AGPS server. These AGPS servers download the orbital information from the satellite and store it in the database. An AGPS capable device can connect to these servers and download this information using Mobile Network radio bearers such as GSM, CDMA, WCDMA, LTE or even using other wireless radio bearers such as Wi-Fi. Usually the data rate of these bearers is high, hence downloading orbital information takes less time.
If your phone's A-GPS cache is corrupt or stale, forcing a refresh can be beneficial. Normally, Android does in fact take care of maintaining this cache automatically. I've personally only used this app to help diagnose and solve GPS problems, something it does very well. In normal usage I wouldn't bother with the A-GPS cache until you find it needs intervention.