Socket Error Exceptions in Python when Scraping

I'm guessing your main problem was that you - without any sleep what so ever - queried the site for a huge amount of invalid urls (you create 3 urls for the years 2015-2017 for 22880 pitchers in total, but most of these do not fall within that scope so you have tens of thousands of queries that return errors).

I'm surprised your IP wasn't banned by site admin. That said: It would be better to do some filtering so you avoid all those error queries...

The filter I applied is not perfect. It checks if the years in the list either appears in the start or end the years given on the site (e.g. '2004 - 2015'). This also creates error links but no way near the amount the original script did.

In code it could look like this:

from urllib.request import urlopen
from bs4 import BeautifulSoup
from time import sleep
import csv

base_url = 'http://www.fangraphs.com/' 
years = ['2017','2016','2015'] 

# Getting Links for letters
letter_links = [] 
data = urlopen('http://www.fangraphs.com/players.aspx')
soup = BeautifulSoup(data, "html.parser") 
for link in soup.find_all('a'):
    try:
        link = base_url + link['href']
        if 'players.aspx?letter=' in link:
            letter_links.append(link)
    except:
        pass
print("[*] Retrieved {} links. Now fetching content for each...".format(len(letter_links)))


# the data resides in two different base_urls:
splits_url = 'http://www.fangraphs.com/statsplits.aspx?'
game_logs_url = 'http://www.fangraphs.com/statsd.aspx?'

# we need (for some reason) players in two lists - pitchers_split and pitchers_game_log - and the rest of the players in two different, pos_players_split and pis_players_game_log
pos_players_split = []
pos_players_game_log = []
pitchers_split = []
pitchers_game_log = []

# and if we wanted to do something with the data from the letter_queries, lets put that in a list for safe keeping:
ind_player_urls = []  
current_letter_count = 0
for link in letter_links:
    current_letter_count +=1
    data = urlopen(link)
    soup = BeautifulSoup(data, "html.parser") 
    trs = soup.find('div', class_='search').find_all('tr')
    for player in trs:
        player_data = [tr.text for tr in player.find_all('td')]
        # To prevent tons of queries to fangraph with invalid years - check if elements from years list exist with the player stat:
        if any(year in player_data[1] for year in years if player_data[1].startswith(year) or player_data[1].endswith(year)):
            href = player.a['href']
            player_data.append(base_url + href)
            # player_data now looks like this:
            # ['David Aardsma', '2004 - 2015', 'P', 'http://www.fangraphs.com/statss.aspx?playerid=1902&position=P']
            ind_player_urls.append(player_data)
            # build the links for game_log and split
            for year in years:
                split = '{}{}&season={}'.format(splits_url,href[12:],year)
                game_log = '{}{}&type=&gds=&gde=&season={}'.format(game_logs_url, href[12:], year)            
                # checking if the player is pitcher or not. We're append both link and name (player_data[0]), so we don't need to extract name later on
                if 'P' in player_data[2]:
                    pitchers_split.append([player_data[0],split])
                    pitchers_game_log.append([player_data[0],game_log])
                else:
                    pos_players_split.append([player_data[0],split])
                    pos_players_game_log.append([player_data[0],game_log])               

    print("[*] Done extracting data for players for letter {} out of {}".format(current_letter_count, len(letter_links)))
    sleep(2)
    # CONSIDER INSERTING CSV-PART HERE....


# Extracting and writing pitcher data to file
with open('SplitsPitchingData2.csv', 'a') as fp:
    writer = csv.writer(fp)
    for i in pitchers_split:
        try:
            row_sp = []
            rows_sp = []
            # all elements in the pitchers_split are lists. Player name is i[1] 
            data = urlopen(i[1])
            soup = BeautifulSoup(data, "html.parser")
            # append name to row_sp from pitchers_split
            row_sp.append(i[0])
            # the page has 3 tables with the class rgMasterTable, the first i Standard, the second Advanced, the 3rd Batted Ball
            # we're only grabbing standard
            table_standard = soup.find_all('table', {"class":"rgMasterTable"})[0]
            trs = table_standard.find_all('tr')
            for tr in trs:
                td = tr.find_all('td')
                for content in td:
                    row_sp.append(content.get_text())
            rows_sp.append(row_sp)
            writer.writerows(rows_sp)       
            sleep(2)
        except Exception as e:
            print(e)
            pass

Since I'm not sure precisely how you wanted the data formatted on output you need some work on that.

If you want to avoid waiting for all letter_links to be extracted before you retrieve the actual pitcher stats (and fine tune your output) you can move the csv writer part up, so it runs as a part of the letter loop. If you do this don't forget to empty the pitchers_split list before grabbing another letter_link...