Manpower is the numeric value represented by the icon with 3 people. It determines how many soldiers you have on duty and in reserve. If you played the games from Europa Universalis series (or similar) you can quickly see the similarities. The Manpower is required to operate your ships as its crew, they are the core of the defense and invasion forces. Ships with a full crew do better during space battles. If you are out of Manpower you can't build new ships and you can't replenish your garrisons in systems or make invasions. Even if you don't have any military goals you need a certain amount of Manpower - they are necessary in producing scout or colonizing ships. When a battle breaks out the Manpower is used to make squads. You don't need to create separate units that are used for invasions. If you see a character with a planet/system icon then this means that this is a system garrison and a space ship icon means that this is a ship crew.
On the Military Status screen, in the side panel, you can see how many Manpower you currently have and its maximum number (1). Below, you can see how much Manpower is used to resupply fleets or systems. The maximum number increases with every system owned. The third value is your net increase. The Manpower value is increased by producing Food on your planets, you can see the values from every system (2). In each system, under FIDS indicators, you can check the garrison's status presented in percentage and the number of armies that are stationing there (3). If a system needs to resupply a garrison you can notice a half-transparent ring around that system which gets filled by taking the Manpower from the overall pool that you currently have (4). Manpower simply appear in your territory, they don't have to be transported by ships like migrants. However, blockades and invasions stops all resupplies. Fleets can resupply their Manpower a little bit during a turn but this process is much quicker in a friendly system.
You can use a special project called Chain Gang Program that "consumes" one population unit in a system and converts it to 300 Manpower units. This project is useful in crisis situations and in overpopulated systems with a high Food production. Thanks to that you can quickly regain a lost citizen without an impact on that system's economy. This also allows you to quickly get another system improvement which is called Exotic Rations. It expands a system's garrison and grants you an increase in Manpower from Food produced in a system. Besides that there are other technologies that increases the number of garrisons, their resistance during fights, increases Manpower production, speeds up resupplies in fleets/systems or increases the number of troops on ships by using particular modules.
The number of soldiers on a planet or in fleet is a certain amount of Manpower. To make 1 ground troop you need 5 Manpower, 1 armored unit requires 15 Manpower and 1 aerial unit costs 30 Manpower. If your army consists of only ground soldiers and your fleet has 300 sailors in total then your military power when attacking a system is 60 soldiers. If you change your army to 50% ground units and 50% armored vehicles then, during an invasion, the same fleet will have 30 soldiers and 10 armored units. Remember that some weapon modules can kill crews during space battles. In the example above you can see a system that starts with a garrison with 200 Manpower so it has 40 soldiers (1). However, a fleet that has one ship and its crew consists of 100 Manpower is worth 20 soldiers during an invasion.