Muslims in different societies use the term "descendants of apes and pigs" for Jews because of the following verses in the Quran:
(162) Ask them concerning the town standing close by the sea. Behold!
they transgressed in the matter of the Sabbath. For on the day of
their Sabbath their fish did come to them openly holding up their
heads but on the day they had no Sabbath, they came not: thus did We
make a trial of them, for they were given to transgression. (163) When
some of them said: "why do ye preach to a people whom Allah will
destroy or visit with a terrible punishment?"― Said the preachers: "To
discharge our duty to your Lord and perchance they may fear Him."
(164) When they disregarded the warnings that had been given them, we
rescued those who forbade evil; but We visited the wrong-doers with a
grievous punishment, because they were given to transgression. (165)
When in their insolence they transgressed (all) prohibition, We said
to them: "Be ye apes, despised and rejected." (166) Behold! thy Lord
did declare that He would send against them, to the Day of Judgment,
those who would afflict them with grievous penalty. Thy Lord is quick
in retribution, but He is also Oft-Forgiving, Most Merciful. Quran 7:162-166 (Read the rest of this chapter to get more context, if you'd like.)
(64) And well ye knew those amongst you who transgressed in the matter
of the Sabbath; We said to them: "Be ye apes despised and rejected."
(65) So We made it an example to their own time and to their
posterity, and a lesson to those who fear Allah. Quran 2:64-65
(60) Say: "Shall I point out to you something much worse than this,
(as judged) by the treatment it received from Allah? Those who
incurred the curse of Allah and His wrath, those of whom some He
transformed into apes and swine, those who worshipped Evil;― these are
(many times) worse in rank and far more astray from the even Path!
To be fair to the Quran, it is not really saying that the Jews at present are direct descendants of apes and pigs. It is just generalizing that Jews that didn't keep the Sabbath were turned into apes and pigs as a warning to others. In Muslim societies, the words "descendants of apes and pigs" is in direct reference to these verses in the Quran. It means that Jews are people who had apes and pigs for their ancestors.
As for whether it is the duty of Muslims to kill the Jews. The verses regarding that in the Quran are:
(29) Fight those who believe not in Allah nor the Last Day, nor hold
that forbidden which hath been forbidden by Allah and His Messenger
nor, acknowledge the Religion of Truth from among the People of the
Book, until they pay the Jizyah with willing submission and feel
themselves subdued. (29) The Jews call Uzayr a son of Allah and the
Christians call Christ the son of Allah. That is a saying from their
mouths; (in this) they but imitate what the Unbelievers of old used to
say. Allah's curse be on them: how they are deluded away from the
Truth! Quran 9: 29-30 (In Arabic, the word used for "fight" is
"Qatilu" which can also mean "Kill" or "Murder". Jizya is a second class citizen's tax. It is still a reality for the Jews in Yemen.
O ye who believe! take not the Jews and the Christians for your
friends and protectors: they are but friends and protectors to each
other. And he amongst you that turns to them (for friendship) is of
them. Verily Allah guideth not a people unjust. (51) Quran 5:51.
Strongest among men in enmity to the Believers wilt thou find the Jews
and Pagans; and nearest among them in love to the Believers wilt thou
find those who say: "We are Christians:" because amongst these are men
devoted to learning and men who have renounced the world, and they are
not arrogant. (82) Quran 5:82. With regards to Christians this verse
seems to contradict the one above, but with regards to Jews, it is
still full of hatred.
Those are some of the verses from the Quran. You have to read the whole Quran (where about a third of the verses are dedicated to cursing the Jews). The above quotes are all from Yusuf Ali's translation, the most moderate English language translation available. Yusuf Ali translated the Quran specifically for the Western audience.
Now I will write from the Hadiths (or Ahadith; sayings of Mohammed and the record of what he did).
Sahih Bukhari, Vol. 4, Book 52, Number 176
Narrated 'Abdullah bin 'Umar:
Allah's Apostle said, "You (i.e. Muslims) will fight wi the Jews till
some of them will hide behind stones. The stones will (betray them)
saying, 'O 'Abdullah (i.e. slave of Allah)! There is a Jew hiding
behind me; so kill him.' "
Sahih Bukhari, Vol. 4, Book 52, Number 177:
Narrated Abu Huraira:
Allah's Apostle said, "The Hour will not be established until you
fight with the Jews, and the stone behind which a Jew will be hiding
will say. "O Muslim! There is a Jew hiding behind me, so kill him."
There are many other Ahadith, but I posted the above two as an example to corroborate what the information in the link that you provided above says. Also Sahih Bukhari is the most authoritative Hadith collection in Sunni Islam, which is followed by 90% of the Muslims in the world.
Hope this answer helps you form your opinion. Feel free to ask more questions if something is not clear.