The speaker's platform below the Capitoline in the Roman Forum Hill was originally a tribunal (a place from which Tribunes addressed the Plebians), but already in ancient times the tribunal became known as the "Rostrum". That name and our own word for a speaker's platform are taken from a Latin word, rostrum -- a ship's prow.
Rostra (the plural, both in Latin and English) were the rams of Mediterranean fighting ships. They were originally made of hard wood, but bronze sheathing or castings over extensions of the ships' keels soon proved more efficient. The strategy was simple: ram a hole in your enemy's ship before he did the same to you. Discipline among the rowers was all-important -- I you didn't back out of that hole before your enemy sunk, both ships would go to the bottom.
When you won a naval victory, you cut off the enemy rostra, brought them home, and mounted them on new ships of your own. The bronze prow was the most costly part of the ship, so bringing home the other guy's rostra meant that you could expand your fleet. After really great victories, you could do some serious bragging by mounting some of the rostra for public display rather than reusing them -- "We beat them so bad we have rostra to spare!" So Duilius mounted six "spare" captured rostra on the face of the forum tribunal after his fleet demolished the Latin fleet at Antium in 338 BC. In 260 BC, C. Duilius Nepos added several more after beating the Carthaginians at Mylae. Julius Caesar's plan to expand what by then was called "the Rostrum" was carried out by Augustus, but Agustus's rostra were decorative fakes -- real bronze showing wealth, but not from enemy ships. (The brick Rostrum seen today is a modern restoration.)
Augustus mounted real rostra from the naval battle of Actium on the front platform of the temple that he built to honor his assassinated uncle, "The Divine Julius (Caesar)", and that was particularly significant for two reasons. First, the Actium rostra were Roman, not foreign: they came from Marc Anthony's fleet. Second, Augustus was proclaiming the dominance and divinity of what was to become the Julio-Claudian dynasty both by putting a temple to his uncle Julius in the forum and by placing trophies on its front. He thus established a new Julio-Claudian Rostrum, and during that dynasty you had to be brave or silly to use the old one. The Flavian Emperors took over after Nero ended the Julio-Claudian line in disgrace, and use of the original Rostrum at the other end of the forum resumed.
Good pictures of Augustus's Rostrum today (modern restoration):
1. Slots show where rostra were mounted:
2. Five small pillars in the center-rear of the Rostrum platform were at the front of the pre-Augustan Rostrum:
An artist's reconstruction (pre-Augustan):
Temple of the Divine Julius: