As Richard said in his answer, the first three lines are given in both English and Klingon. These three lines are translated and explained in The Klingon Way by Marc Okrand on p. 203:
DaHjaj SuvwI''e' jIH. = Today I am a warrior.
tIqwIj Sa'angnIS. = I must show you my heart.
'Iw bIQtIqDaq jIjaH. = I travel the river of blood.
These phrases are accompanied by this explanation:
Upon reaching a certain age, the Age of Ascension, a young Klingon
undergoes a rite of passage symbolizing the attainment of a certain
spiritual level. After intoning the three ancient and sacred phrases
recorded above, the initiate, while walking along a path lined by
painstik-wielding warriors, expresses his or her deepest feelings.
Marc Okrand is the inventor of the Klingon language and was a language consultant on many (though not all) of the Star Trek movies and episodes where Klingon was used. The above sentences are not only not gibberish, but are in fact real grammatical sentences comprehensible to anyone who has read the 2nd edition of The Klingon Dictionary. (The 2nd edition was amended to include words used in TNG and Star Trek VI.)
The first sentence breaks down as follows:
DaHjaj = today (DaH = now, jaj = day)
SuvwI' = warrior (Suv = fight, -wI' = one who does)
-'e' = topic marker
jIH = I am
The second sentence uses:
tIq = heart
-wIj = my
Sa- = verb prefix meaning I-you (plural)
'ang = show
-nIS = need, must
As I noted in the other comment, there are two typos in the linked transcripts: tIq ends in "q" not "g", and -nIS has a lower-case "n". The transcription system devised by Okrand for Klingon is case-sensitive, and those are fairly obvious typos made by an English-speaker unfamiliar with the system. (In English, "q" is always followed by "u", so an English speaker can easily misread "t-I-q" as "t-I-g". Also, someone can hit the shift key too early when typing "-n-I-S", if they didn't understand that "n" must be lower-cased while "I" must be upper-cased.)
Finally, the third sentence has:
'Iw = blood
bIQtIq = river
-Daq = locative suffix
jI- = verb prefix meaning I
jaH = go, travel
So that's that. As for the sentences which were left untranslated in the episode:
jI- = verb prefix meaning I (same as above)
bech = suffer
-rup = be ready or prepared to
Thus, jIbechrup means "I am ready to suffer".
may' = battle
vI- = verb prefix meaning I-it
loS = await, wait for
Thus, may' vIloS means "I await battle".
HI- = imperative prefix, ordering someone to do something to me
HIv = attack
-qa' = suffix meaning to resume, do again
Thus, HIHIvqa' means "Attack me again!" Note that Worf gives this command to the final pair of Klingons.
As with the other sentences, these three sentences use only words from The Klingon Dictionary, and would be readily understood by anyone who's familiar with the content of that book.
I don't know why they (the writers or producers or whoever) decided to give both Klingon and English versions for only the first three sentences but not the rest. Perhaps they didn't want the audience to be completely lost if they didn't understand Klingon, but still wanted to provide an air of authenticity. Perhaps memorising so many lines in Klingon was too much for Michael Dorn. But it seems that Marc Okrand did translate these sentences into Klingon also, since the vocabulary exists to express them. In particular, the expression involving the "bile of the vanquished" appears in a slightly altered form in Power Klingon, where it is given as a wish spoken to someone who had just undergone the Rite of Ascension:
jagh lucharghlu'ta'bogh HuH ghopDu'lIj lungaSjaj. = May the bile of the vanquished fill your hands.
The last Klingon sentence spoken in Worf's ceremony, given in the transcript as May'pequ' moH, is a bit tricky.
may' = battle (as above)
pe- = imperative prefix, ordering multiple people to do something (no object)
qu' = be fierce; or
-qu' = emphatic suffix
-moH = causative suffix
The sentence, as written, doesn't seem to make sense. Fixing the casing and spacing, it would be may' pequ'moH. Let's start from the back. The causative suffix -moH is perhaps best explained by example. If you had the verb jaq meaning "be bold", then jaqmoH means "embolden". The verb pum means "to fall", so pummoH means "to knock down". So qu'moH would mean something like "en-fierc-en" or "fierc-ify", i.e., to cause someone or something to be fierce. There's no easy way to express this concept in English, but it's just qu'moH in Klingon.
The problem, though, is that the prefix pe- indicates no object (i.e., it is an order for multiple people to do something, but not do it to anyone or anything in particular). And yet the sentence has a word in the object position, namely, may' "battle". Since the punctuation seemed to have been dropped between jIbechrup and may' vIloS, perhaps whoever copied this sentence into the script also dropped the punctuation (or more) here, and it's supposed to be two sentences. (They definitely messed up the spaces.) It's also possible that a verb is missing, since -qu' is also a verb suffix.
But, as given, I'd interpret the sentence as:
may'! pequ'moH! = Battle! Make fierce! (The latter is an order given to multiple people, presumably the Klingon warriors. What they are supposed to make fierce is left unspecified.)
In addition to the above listed sources, I am a Klingon speaker and long-time member of the Klingon Language Institute.