The phrase all roads lead to Rome, and its variants, mean: there are many different ways of reaching the same goal or conclusion.

This phrase is probably ultimately after post-classical Latin mille viae ducunt homines per saecula Romam…, i.e., a thousand roads lead for ever to Rome the men…, which occurred in Liber Parabolarum (c.1175), by the French theologian and poet Alain de Lille (Latin: Alanus ab Insulis – born c.1128–died 1202 or 1203). This Latin phrase is as follows in Liber Parabolarum Ālani (Argentoratum, 1513)—source: Bayerische Staatsbibliothek (Bavarian State Library):

Mille viæ ducũt homines ꝑ sæcula romam:
Qui dominum toto querere corde volunt.
     translation:
A thousand roads lead for ever to Rome the men
Who desire to seek the Lord with all their heart.

The metaphor occurred in the prologue to A Treatise on the Astrolabe (c.1391), by the English poet Geoffrey Chaucer (c.1340–1400)—source: Internet History Sourcebooks Project, Fordham University:

This tretis, divided in 5 parties, wol I shewe the under full light reules and naked wordes in Englissh, for Latyn ne canst thou yit but small, my litel sone. But natheles suffise to the these trewe conclusions in Englissh as wel as sufficith to these noble clerkes Grekes these same conclusions in Grek; and to Arabiens in Arabik, and to Jewes in Ebrew, and to the Latyn folk in Latyn; whiche Latyn folk had hem first out of othere dyverse langages, and writen hem in her owne tunge, that is to seyn, in Latyn. And God woot that in alle these langages and in many moo han these conclusions ben suffisantly lerned and taught, and yit by diverse reules; right as diverse pathes leden diverse folk the righte way to Rome.

The earliest occurrences of the phrase all roads lead to Rome and variants that I have found are as follows, in chronological order:

1-: From a letter, dated 15^th March 1793, that M. Ehrenstrom wrote to Baron Armfelt, published in The Correspondence of Baron Armfelt and the other conspirators against the Swedish Government: Translated from the Swedish copy published by the Government at Stockholm (London: Printed for G. G. and J. Robinson, 1795):

This difference in the choice of our means ought not to stop our career. All roads lead to Rome. We have only to choose the surest way of arriving at the happiest issue.

2-: From The conciliating Lawyer, the sick man’s Friend, and the Hermit, published in La Fontaine’s Fables. Now first translated from the French, by Robert Thomson (Paris: Sold by Chenu, 1806):

Three diff’rent roads the three concurrents chose,
All roads alike conduct to Rome.

The original passage is as follows in Le Juge Arbitre, l’Hospitalier, & le Solitaire, as published in Fables choisies. Par M^r De la Fontaine (Paris: Chez Claude Barbin, 1694), by the French poet Jean de La Fontaine (1621–1695):

Ils s’y prirent tous trois par des routes diverses.
Tous chemins vont à Rome.

3-: From General reflections suggested by Italy, seen in the years 1818 and 1819, published in The London Magazine (London: Printed for Baldwin, Cradock, and Joy) of January 1820:

The traveller is now in the country of the eternal city!—she that, by the universal confession, is the common centre of the civilized world. That all the roads of the earth lead to Rome, was said of her in the days of her military supremacy; and it was continued to be said when, deprived of all the common constituents of force, she nevertheless exercised an absolute power over the most powerful monarchs.

4-: From The Brothers; Or, The Castle of Niolo. A Romance (London: Published by William Emans, 1820), by the English author Robert Huish (1777–1850):

With the greatest caution and silence, Rosenheim and Rupert followed the abbot; they observed him at times turn suddenly round, and stop for a few minutes, as if to discover if he were watched. They then concealed themselves behind some bush or some jutting rock—nor did they emerge from their concealment, until the abbot pursued his course. He had now reached the hermitage, and standing for some time at the door, he entered it.
“What can the reprobate have to do there,” said Rupert,” that is not the way to the monastery.”
“There are more ways lead to Rome,” said Rosenheim, “than you are aware of—come, let us hasten, or we shall miss him.”

5.1 & 5.2-: From Lessons of Thrift. Published for General Benefit, by a Member of the Save-all Club (London: Printed for Thomas Boys, 1820):

5.1-: From Lesson XVI. Money lent, money lost:

The furthest way about is the nearest way home. There are many roads that lead to Rome. These are proverbs, but proverbs are the traditions of ancient wisdom.

5.2-: From To Mr. Merryweather, Secretary of the Save-all Club, by ‘P. Ironsides’:

As a little corner of my bit of paper is left, I shall give or lend you a few hints, not in the style of the passionate man, who kicked an insolent fellow down stairs, and said he had given him a gentle hint. No, no. I am your friend, and love your plan, however imperfect. There are many roads that lead to Rome, and Rome was not built in one day.

On the pattern of all roads lead to Rome, the phrase all roads lead to —— means: all ways lead ultimately to (the place, topic, conclusion, etc., specified as being the most central, dominant or important).

For example, the following is from Visit to the London Times printing-office, published in The Anglo American: A Journal of Literature, News, Politics, the Drama, Fine Arts, etc. (New York: E. L. Garvin & Co.) of Saturday 2^nd September 1843:

At Paris there is an establishment for the Times reporters, each morning paper possessing a similar one, which, besides supplying articles of news regarding the French capital, forms a sort of agency for the management of expresses overland from India, and various other parts of the world; as if to confirm the French boast, that “all roads lead to Paris.”