The names of Polish months are unique in the way that Poland did not adopt Roman names - like many other countries in Europe (France, Germany, Russia). Original, non-Latin names for the months of the year also exist in some other languages - like Czech (but not Slovak), Croatian, Lithuanian or Basque. But this is rather an exception than the rule.
Check the website with the names for the months of the year in several languages. Check also another article devoted to the origin of the days of the week in Poland.
It is worth to mention that even in the cultures that adopted Roman names there is an effort to recreate original pre-Roman names - like in Russia or Germany. For instance according to the old German language the name for January - in German Januar has its own German name - Wintermanoth or Hartung.
Refrerring to the Roman calendar, the names of the months derive either from the Roman names of the gods: (January - June), from the names of the ceasars: Julius Ceasar and Augustus Caesar (July-August). The names from September to December originate from the Latin successive numbers for a given month which is a bit confusing since September comes from Latin "septem" meaning seventh since September was originally the seventh month of the year and the year originally started in March and lasted only 10 months. Two winter months were unnamed since they were not important in agriculture. October means the eight (octo), November - the ninth (novem) and December - the tenth [month].
Now, let me focus on the meaning of the months in Polish language, tradition and history. The months names are related to the farmers calendar through the year.
Styczeń - stykac means "to meet" - "to join" since he old year "meets: the new year in January
Luty - luty in old Polish means "fierce", "bitter frost", "freezing cold"
Marzec - from marznac meaning " to freeze". Some people believe that this name comes also from Roman God "Martius" - in Polish Mars.
Kwiecień - from kwiecie - flowers - a blooming month.
Maj - this is the only name which was adopted from Roman calendar devoted to goddess Maia. Maia was the Roman female deity of growth or fertility.This name gained its own meaning in the Polish language, for instance as majowka- the outdoor trip.
Czerwiec - from czerw meaning "grab" - the larve of a bee or moth. In this month people were picking up the larves dry them in the sun and then made from it a red or purple die (pigment); therefore the name "czerw" comes from "redden" and some sources also derive the name of Polish June from a Polish word "czerwienic" - to redden or rippen
Lipiec - from lipa - "linden tree" - which flowers in that time; this tree is very popular in Poland. There is a famous poem by Kochanowski "Na lipe" (about Linden tree) - "gosciu siadz pod ma lipa a odpoczyn sobie" - "dear visitor -please sit under my linden tree and rest" . Read Love and Lore of the Linden
Sierpień - sierp is "a sickle" used for harvesting (tool to cut the hay, grass or wheat)
Wrzesień - from wrzosy - "heather" that beautifully purple in that time of the year
Październik - from paździerz - tow, wooden dry part remained from flax or hemp. In the past flax and hemp was used for making cloths. In that time of the year the wind was carrying out "paździerze" - the wooden waste remained after flax and hemp - all over the fields.
Listopad - padajace liscie - falling leaves.
Grudzień - gruda - hardened ground which is caused by cold weather.
It is worth to mention that some of the months names are similar in common to several Slavic lanuages for instance "listopad" which is 11th month (November is Poland) has the same name for October in Croatian and Slovenian and November in Czech. The most similarities are between Polish and Czech languages. In Czech "kveten" is the fifth month but its name resembles Polish April's name - "kwiecien", "cerven" for Czech Mai is similar to Polish "czerwiec" for June and "srpen" Polish August's name sierpien.
By the way, Polish names of the months are written not with capital letters like in English. But some months names in Czech are very different from Polish - for instance December is called "prosinec" - month of hog-killing (originally prasinec, prase = hog). In this aspect Polish name for December, grudzien, is similar rather to Ukrainian: gruden.