When Islam is against idol worship why do the Muslims worship, and
bow down to the Kaaba in their prayer?
Kaaba is the Qibla i.e. the direction Muslims face during their prayers.
It is important to note that though Muslims face the Kaaba during prayers,
they do not worship the Kaaba. Muslims worship and bow to none but Allah
.
It is mentioned in Surah Baqarah:
We see the turning of thy face (for guidance) to the heavens: now shall
We turn thee to a Qiblah that shall please thee. Turn then thy face in the
direction of the Sacred Mosque: wherever ye are, turn your faces in that
direction.
Al-Qur'an Surah Al-Baqarah 2:144
Islam believes in fostering unity
For instance, if Muslims want to offer Salaah (Prayer), it is possible
that some may wish to face north, while some may wish to face south. In
order to unite Muslims in their worship of the One True God, Muslims,
wherever they may be, are asked to face in only one direction i.e. towards
the Kaaba. If some Muslims live towards the west of the Kaaba they face
the east. Similarly if they live towards the east of the Kaaba they face
the west.
Kaaba is at the centre of the world map
The Muslims were the first people to draw the map of the world. They
drew the map with the south facing upwards and north downwards. The Kaaba was
at the centre. Later, western cartographers drew the map upside down with
the north facing upwards and south downwards. Yet, Alhamdullilah the Kaaba
is at the centre of the world map.
Tawaaf around Kaaba for indicating one God
When the Muslims go to Masjid-e-Haram in Makkah, they perform tawaaf or
circumambulation round the Kaaba. This act symbolizes the belief and worship
of One God, since, just as every circle has one centre, so also there is
only one Allah [swt] worthy of worship.
Hadith of Umar 
Regarding the black stone, hajr-e-aswad, there is a hadith (tradition),
attributed to the illustrious companion of the Prophet Muhammed 
,
Umar .
According to Sahih Bukhari, Volume 2, book of Hajj, chapter 56,
Hadith No. 675. Umar said, "I know that you are a stone and can
neither benefit nor harm. Had I not seen the Prophet touching (and
kissing) you, I would never have touched (and kissed) you".
People stood on Kaaba and gave the adhaan
At the time of the Prophet , people even stood on the Kaaba and gave
the 'adhaan' or the call to prayer. One may ask those who allege that
Muslims worship the Kaaba; which idol worshipper stands on the idol he
worships?