Heroes are leveled up using XP Meta-Agents. These can be acquired through normal gameplay, including (but not limited to) completing Campaign Missions and taking part in select Upgrade Events.
On the "Heroes" page, tap the icon of a Hero in your collection to view their profile. Tap the Level Up button to view your collection of XP Meta-Agents.
From here, tap on the XP Meta-Agent that you wish to apply. If you have more than one of that type of Meta-Agent, you will be prompted to select the amount of Meta-Agents of that type that you wish to apply. If you prefer to apply all of the XP Meta-Agents that you have, simply tap the "Auto-Select" button.
Once you have made your selections, tap the "Apply" button. Note that there is a cost in green Essence, based on the amount of XP Meta-Agents applied and the amount of XP that your Hero is due to receive. You can deselect XP Meta-Agents at any point before confirming your selections and paying the green Essence cost.
Please note, your Hero's level cannot exceed your current Account Level. If you are close to your Hero's max level based on your current Account Level, then you will only apply the XP Meta-Agents needed to reach that max level. Excess Meta-Agents would not be spent or loss in this case, meaning you can save those and/or apply them to another Hero in your collection.