Many of the capacitors serve as decoupling for various ICs on the board. When a logic line switches states, it is fighting parasitic capacitance of the PCB traces and whatever load that line is driving, as well as parasitic capacitance internal to the chips themselves. The effect is that the chips tend to draw small surges of current each time anything switches states, which obviously happens many many times a second. Without these decoupling capacitors, the resistance of the traces feeding power to the different parts will fluctuate with each of these surges and the parasitic inductance can cause unwanted resonance, leading to a great deal of random noise riding on the power rails. The capacitors serve as localized storage to absorb these spikes, "decoupling" them from other parts of the circuit. It can be thought of as similar to water towers located in various places around town. They help maintain a steady water pressure at the individual neighborhoods despite variations in demand by creating a local reservoir, isolating the area from pressure variations in the main line.
The other common application of those problematic SMT electrolytics is coupling and DC blocking in the audio amplifier circuit. The audio signal is passed through a capacitor, which passes AC but blocks DC, removing the DC offset and resulting in an AC audio signal. When these fail, the audio becomes soft or vanishes.
The other common application of those problematic SMT electrolytics is coupling and DC blocking in the audio amplifier circuit. The audio signal is passed through a capacitor, which passes AC but blocks DC, removing the DC offset and resulting in an AC audio signal. When these fail, the audio becomes soft or vanishes.