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To begin, what are base stats?
They are the stats that a champion has at level 0 and gains a certain amount per level. For example, Maokai has 58 base AD at level 0 and he gains 3.3AD per level. Therefore, the amount shown in game at lvl 1 would be 61 AD. The exception for gaining per level is movespeed and Magic resistance for ranged characters.
Some such as movement speed, HP,etc. can be modified with items. Some cannot be modified except with abilities such as attack range. A champions base AD and AS cannot be modified by items or buffs, these simply add "bonus AD" or more AS. (Look at how Sheen works with base AD). Also a champion's base AS affects the champion when buying additional attack speed items, due to the formula.
Some other stats are not part of base stats but are only acquired through items or other means. This includes crit damage, armor pen, ability power, etc.
How do base stats affect gameplay?
Lets look at Sona. Her base hp was nerfed to be the lowest of any champion due to her "free" power. Now that her passive armor is nerfed, her base hp can be higher. It's all in balance, if a champion has too much zone control, their base stats will be low to compensate. Zyra, Lux and Anivia have great zone control and are among the lowest in armor and base hp, they also have low movement speed. Another issue is free utility, Janna, Soraka and Nami have free utility and are ranged, therefore have low base stats.
On the other end, melee champions need to be sturdy, so they get high base hp and AD, fighters without gap closers also usually have high base movement speed. This includes tanks such as Amumu, Zac and Alistar. But also most fighters have at least 340ms and magic resistance per level to compete with ranged champions and because they will be in the middle of the fight due to being melee.
Strengths and weaknesses
Now if you compare Thresh's base hp and armor with someone like Janna or Sona, you can clearly see a big difference. This is meant because Thresh is a "tanky disrupting support". But he is ranged and has an autoattack steroid as well. Might as well pick him for better base stats and similar ranged harass! Although he is getting nerfs soon for this reason.
Here is where lies the balance in picking champions and knowing matchups. If you are Riven and play against Lux or Heimer top, who have some of the lowest base hp, you should definitely all-in them at lvl 2-3 because you will most likely come out on top due to base stats alone. On the contrary, should you face someone such as Jax, Gangplank or Tryndamere with highest base hp, be careful about all-inning them with a champ with weaker stats.
Karthus has the lowest lvl 1 AD, at 45. Maokai, Morde, Shaco, Akali and Olaf have the highest base AS. Before the recent changes, Kayle and Nidalee were the only champions to gain 0.75 magic resist per level, due to them being "sometimes ranged". Nunu no longer has the highest hp at lvl 18, it is now Alistar. So next time you play, compare your level 1 hp and other stats with your allies, and your enemy laners, this can help you in very subtle ways!