Bharata was a hero because he was endowed with heroism, that is, the ability to conquer one’s [spiritual] enemies such as passion. In other words, His mind was not attracted to sense objects.

ebhiś ca sacivaiḥ sārdhaṁ śirasā yācito mayā
bhrātuḥ śiṣyasya dāsasya prasādaṁ kartum arhasi

“With these assistants, I bow down My head and beg You: You should have mercy upon Me, Your brother, disciple and servant.” (Rāmāyaṇa 2.104.12)

From the above verse, it is clear that Bharata tried to please the lotus feet of Rāma with His fourfold army. The expression “feet” is a sign of being worshipable—pūjye tu pāda-nāmāṅkaḥ: “A name [of a person] suffixed with pāda indicates that he is worshipable.” (Amara-śeṣa) In other words, Bharata wanted to please His worshipable Rāma. Because Bharata was a servant of Lord Rāma, He considered that He was trying to please the lotus feet of Rāma. Such is the usage and convention of servants. Indeed, the servants of a king say that they are going to see the feet of the king instead of saying that they are going to see the king.

Rāma was not angry at Bharata, but Bharata wanted to remove Rāma’s conception that the kingdom had been given to Him. Bharata would also relate this later on:

pūjitā māmikā mātā dattaṁ rājyam idaṁ mama
tad dadāmi punas tubhyaṁ yathā tvam adadā mama

“My mother has been honored [by You]. This kingdom was [merely] entrusted to Me [by You]. I give it back to You just as You had given it to Me.” (Rāmāyaṇa 6.131.2)