My understanding is that the quote comes from Wikipedia article on Bakunin (also quoted verbatim on Marxist internet archive.) The reason why I bothered to check the source is because the quote contains a false statement about Marx:
[...] Marx, who was a key figure in the General Council of the International and argued for the use of the state to bring about socialism.
(emphasis is mine)
For Marx the State was always an enemy - a tool by means of which one class dominated the other classes. According to Marx the state was to be overthrown in a proletarian revolution, or at least subjugated to the demands of the proletariat (dictatorship of the proletariat) to eventually wither away with the rise of the Communism. The most annoying (to Marx) proponent of bringing Socialism via the state intervention was Ferdinand Lassalle, who established the first German Labor party with precisely this aim: to exert political influence on the German (monarchical) government, in order to achieve the universal suffrage, creation of the social safety net, and transition cooperative management of industry. He partially succeeded in this, as Bismarck did introduce welfare reforms, although already after the Lassalle's early death (which is also the reason why he wasn't a member of the First international.) The record of the direct one-on-one consultations between Lassalle and Bismarck would be discovered in 1920 and scandalize the Communists. (See more about the Socialism via state intervention in this answer.)
Marx and Bakunin
The first international was a loose union of various socialists and communist movements, which at first ended up to be dominated by Marx (notably after the expulsion of Bakunin), but eventually had to be dissolved when the rank and file members revolted against Marx support for the brutal measures by the Paris Commune (Marxist literature skims over this question, preferring to focus on even more brutal suppression of the Commune.)
It is necessary to note that Marx used terms Socialism and Communism interchangeably and distanced himself from many "classics" of Socialism, such as Saint-Simon, Fourier, Proudhon, etc. (He even viciously attacked the ideas of some of them.) Followers of many of these movements were a part of the First International.
Bakunin is often considered as the father of Anarchism, and is described today as Communist anarchist. His major differences with Marx were on the form of the Communist society and the methods of achieving it:
- Bakunin believed in complete individual freedom and governing via consensus. This is the essence of his anarchism - that humans have propensity for collaboration, rather than purely egoist behavior. Kropotkin, who has significant expertise in biology, would later give more basis to these claims. Marx, on the other hand believed in more centralized government - he essentially envisioned a society run by a Communist party (although in details this was spelled by his followers.)
- Bakunin essentially favored terrorist tactics, which would earn the anarchists bad name. The peak of these was the assassination of Alexander II of Russia in 1881 (after Bakunin's death), but in his lifetime he took part in many resurrections all over Europe. Marx expected a mass rising by proletariat against the bourgeoisie, and necessity of going through all the stages of Capitalism before Communism can be achieved.
Here is a quote about Bakunin from Isaiah Berlin's Karl Marx: His Life and Environment:
On this ground he was passionately opposed to the aim accepted by both Marx and the reformists – the replacement of
the status quo by a centralised socialism – since, according to
him, this was a new form of tyranny at once meaner and more
absolute than the personal and class despotism it was intended
to supplant. This attitude had as its emotional basis a temperamental dislike of ordered forms of life in normal civilised society,
a discipline taken for granted in the ideas of Western democrats,
which to a man of his luxuriant imagination, chaotic habits and
hatred of all restraints and barriers seemed colourless, petty,
oppressive and vulgar. An alliance built on an almost complete
absence of common aims could not last: the orderly, rigid, unimpressionable Marx came to regard Bakunin as half charlatan,
half madman, and his views as absurd and barbarian. He saw in
Bakunin’s doctrine a development of the wild individualism for
which he had already condemned Stirner; but whereas Stirner
was an obscure instructor in a high school for girls, a politically
ineffective intellectual, neither capable nor ambitious of stirring
the masses, Bakunin was a resolute man of action, an adroit and
fearless agitator, a magnificent orator, a dangerous megalomaniac
consumed by a fanatical desire to dominate men, at least intellectually, fully equal to that which possessed Marx himself.
More specifically, the split and the expulsion of Bakunin from the International occurred because Marx tried to transform the International into a disciplined party subjugated to him, while Bakunin, faithful to his anarchist principles, saw it as a union of independent movements of even individuals. Again quoting Berlin:
Marx’s chief concern was to arrive at a clear
formulation of a concrete international policy in terms of spe-
cific demands co-ordinated with each other, and the creation of
a rigorous discipline which guaranteed undeviating adhesion to
this policy. He therefore successfully resisted all offers of alliance
with such purely humanitarian bodies as the League of Peace
and Freedom, then newly founded under the aegis of Mazzini,
Bakunin and Mill.
This dictatorial policy was bound, sooner or later, to lead
to discontent and rebellion; these crystallised round Bakunin,
whose conception of a loose federation of semi-independent local
bodies began to gain adherents in the Swiss and Italian sections
of the International, and to a lesser extent in France. Finally they
resolved to constitute themselves, under Bakunin’s leadership,
into a body to be called the Democratic Alliance, affiliated to
the International, but with an internal organisation of its own,
pledged to resist centralisation and to support federal autonomy.
This was a heresy which even a more tolerant man than Marx
could not afford to overlook: the International was not intended
to be a mere correspondence society between a loose association
of radical committees, but a unified political party pressing for a
single end in all the centres of its dispersion.
The OP adds that
How did they argue for/against the necessity of a transition state, and what role did Bakunin's "secret organisation" play in this?
In this respect the initial quote should be read inn a narrow sense: Marx wanted the transformation towards the Communist society (after the revolution) handled by a transition state (aka dictatorship of the proletariat), while Bakunin objected even this form of state as inherently oppressive (and the Communist regimes of the XX-th century have proven that he had a point.)