Here´s my translation

Alan

Þá mælti bæjarmaðurinn: "Hver er þessi maður?"
Then spoke the-townsman: „Who is this man?“

Kjartan sagði nafn sitt.
Kjartan said his name.

Bæjarmaður mælti: "Þú ert sundfær vel eða ertu að öðrum íþróttum jafn vel
The townsman spoke: “You are well able-to-swim or (so) are you at other sports equally well

búinn sem að þessi?"
equipped as at this (one)?”

Kjartan svarar og heldur seint: "Það var orð á þá er eg var á Íslandi að þar
Kjartan answers and rather slowly: “That was (the) word (report) on (it) then when I was in Iceland that there

færu aðrar eftir. En nú er lítils um þessa vert."
other (sports) went (subjunctive of fara) after (? perhaps accordingly, ie in like manner; or maybe, behind, ie not as good) . But now (there) is little of worth concerning this  (ie that´s nothing to write home about).”

Bæjarmaður mælti: "Það skiptir nokkuru við hvern þú hefir átt eða hví spyrð þú mig engis?"
(The) townsman spoke: “That depends (skipta, Z4) somewhat with whom you have had-to-deal-with (eiga við e-n, Z10) or (so) why do you ask ask me for nothing?”

Kjartan mælti: "Ekki hirði eg um nafn þitt."
Kjartan spoke: “I care (present tense) nothing about your name.”

Bæjarmaður segir: "Bæði er að þú ert gervilegur maður enda lætur þú
(The) townsman says: “Both it is that you are an accomplished person (man) and-moreover you behave

allstórlega. En eigi því síður skaltu vita nafn mitt eða við hvern þú hefir
very-haughtily. But not for that less (none-the-less), you shall know my name or (and) with whom you have

sundið þreytt. Hér er Ólafur konungur Tryggvason."
contended-in the-swimming. Here is Ólafr Tryggvi´s-son.”

Kjartan svarar engu og snýr þegar í brott skikkjulaus. Hann var í
Kjartan says nothing and turns (goes) at once away cloakless. He was in

skarlatskyrtli rauðum. Konungur var þá mjög klæddur. Hann kallar á Kjartan
red kirtle-of-scarlet (cloth). (The) king was then (by that time) much clothed. He calls to Kjartan

og bað hann eigi svo skjótt fara. Kjartan víkur aftur og heldur seint.
and bade (past tense) him not so quickly to go. Kjartan turns back (returns) and rather slowly.

Þá tekur konungur af herðum sér skikkju góða og gaf Kjartani,
Then (the) king takes from his (own) shoulders a fine cloak and gave (it) to Kjartan,

kvað hann eigi skikkjulausan skyldu ganga til sinna manna. Kjartan þakkar konungi gjöfina
declared he (accusative, ie Kjartan) should not walk cloakless to his people (men). Kjartan thanks (the) king for the-gift (fem acc sg + def art suffixed)

og gengur til sinna manna og sýnir þeim skikkjuna. Ekki létu hans menn vel yfir þessu,
and walks to his men and shows them the-cloak. His men did not express approval over this,

þótti Kjartan mjög hafa gengið á konungs vald. Og er nú kyrrt.
thought Kjartan greatly to have gone into (submitted to) (the) kings power. And (it) is now quiet.

Veðráttu gerði harða um haustið. Voru frost mikil og kuldar. Heiðnir menn
(it) made (the) state-of-the weather severe (ie the weather turned nasty, see göra, Z12) during the-autumn. (There) were great frosts (plural) and continued-cold-weather (plural). Heathen folk (men)

segja það eigi undarlegt að veðrátta léti illa: "Geldur að nýbreytni konungs
say that (to be) not extraordinary that (the) state-of-the-weather behaves badly: “(it) repays (it´s payback time) for (the) innovation-of-the-king

og þessa hins nýja siðar er goðin hafa reiðst."
and this the new faith at which the-gods (neut nom pl + def art suffixed) have-become-angry.”

Íslendingar voru allir saman um veturinn í bænum. Var Kjartan mjög fyrir þeim.
(The) Icelanders were (stayed) all together during the-winter in the-town. Kjartan was much before(ie to the fore with, ie he lead) them.

Veðrátta batnar og komu menn fjölmennt þá til bæjarins að orðsending
(The) state-of-the-weather improves and people (men) came in-large-numbers then to the-town at (the) message (behest)

Ólafs konungs. Margir menn höfðu við kristni tekið í Þrándheimi en hinir
of King Ólafr. Many men had accepted Christianity in Trondheim but those (hinn, Z2, emphatic)

voru þó miklu fleiri er í móti voru.
were nevertheless much more who were against.

Einnhvern dag átti konungur þing í bænum út á Eyrum og talaði trú fyrir
One day (the) king had a Thing in the town out on Eyrr (The Spit) and talked (as in preached) (the) faith before

mönnum, langt erindi og snjallt. Þrændir höfðu her manns og buðu konungi
(the) people (men), a long and eloquent speech. (The) Trondites (people of Trondheim) had a man´s crowd (host of men, see herr, Z1) and offered to (the) king

bardaga í mót. Konungur kvað þá vita skyldu að hann þóttist átt hafa við
a battle in return. (The) King declared those (ie that they) should know that he bethought-himself to have had-to-deal with (eiga við, Z10)

meira ofurefli en berjast þar við þorpara í Þrándheimi. Skaut þá bóndum skelk í bringu
greater overwhelming forces than to fight there against cotters (þorpari, peasant-scum) in Trondheim. The farmers were seized with fear

og lögðu allt á konungs vald og var margt fólk þá skírt.
and placed everything in (the) king´s power and many folk were then baptised.

En síðan var slitið þinginu.
But after-that (it) was broken-off in the Thing (ie the Thing was dissolved).