The answer is pretty short and sweet: because he had an army.

It's commonly touted that Antony was Caesar's right-hand-man, but as a matter of fact Lepidus was. Antony had a habit of getting his own foot stuck in his mouth like the time he tried to present Caesar with a crown. After Caesar's death, Lepidus was sent to Spain to make sure that Sextus Pompey would not stir up trouble.

With Lepidus gone, Mark Antony was turning Caesar's assassination into a civil war against the conspirators. The Senate sent the consuls and Octavian against him to nip the war in the bud, but both consuls ended up dying in battle. Octavian lucked into controlling two consular armies along with his own private forces, and Antony had a large army of Caesar's loyalists. Lepidus was called assist, but ended up joining Antony. Octavian joined them too. A common view is that each of them were essentially forced into an agreement by their armies (the soldiers wanted to join Antony's vendetta), but it is also possible that Lepidus and Octavian were making the best of the loss of the consuls.

At any rate, together they controlled the manpower of the western empire and began the triumvirate ostensibly to make peace in the empire so that the republic could run as usual. Lepidus was in control of Spain and particularly had to deal with Sextus Pompey who had started a costly pirate war. Octavian took advantage of Lepidus's difficulties and took more and more of his territory until Lepidus felt forced to fight Octavian and was killed. Antony and Lepidus both saw Octavian of overstepping his power and felt justified when they went to war with him (at separate times), but Octavian was able to present these as unprompted aggression.